Harmonic series: Difference between revisions
Thundergnat (talk | contribs) m (Promote. multiple implementations, little controversy) |
Thundergnat (talk | contribs) m (syntax highlighting fixup automation) |
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=={{header|ALGOL 68}}== |
=={{header|ALGOL 68}}== |
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Using standard lenghth REAL numbers this can find the first Harmonic number > 10, would probably need higher precision to find Harmonic numbers with larger values. |
Using standard lenghth REAL numbers this can find the first Harmonic number > 10, would probably need higher precision to find Harmonic numbers with larger values. |
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< |
<syntaxhighlight lang="algol68">BEGIN # find some harmonic numbers, Hn is the sum if the reciprocals of 1..n # |
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# returns the first n Harmonic numbers # |
# returns the first n Harmonic numbers # |
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OP HARMONIC = ( INT n )[]REAL: |
OP HARMONIC = ( INT n )[]REAL: |
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FI |
FI |
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OD |
OD |
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END</ |
END</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=={{header|Arturo}}== |
=={{header|Arturo}}== |
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< |
<syntaxhighlight lang="rebol">H: function [n][ |
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sum map 1..n => reciprocal |
sum map 1..n => reciprocal |
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] |
] |
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loop 1..4 'l [ |
loop 1..4 'l [ |
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print ["Position of first term >" l ":" firstAbove l] |
print ["Position of first term >" l ":" firstAbove l] |
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]</ |
]</syntaxhighlight> |
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{{out}} |
{{out}} |
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=={{header|AWK}}== |
=={{header|AWK}}== |
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<syntaxhighlight lang="awk"> |
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<lang AWK> |
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# syntax: GAWK -f HARMONIC_SERIES.AWK |
# syntax: GAWK -f HARMONIC_SERIES.AWK |
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# converted from FreeBASIC |
# converted from FreeBASIC |
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exit(0) |
exit(0) |
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} |
} |
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</syntaxhighlight> |
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</lang> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=={{header|BASIC}}== |
=={{header|BASIC}}== |
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==={{header|BASIC256}}=== |
==={{header|BASIC256}}=== |
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< |
<syntaxhighlight lang="freebasic">h = 0.0 |
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print "The first twenty harmonic numbers are:" |
print "The first twenty harmonic numbers are:" |
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print "The first harmonic number greater than "; i; " is "; h; ", at position "; n-1 |
print "The first harmonic number greater than "; i; " is "; h; ", at position "; n-1 |
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next i |
next i |
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end</ |
end</syntaxhighlight> |
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==={{header|QBasic}}=== |
==={{header|QBasic}}=== |
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{{works with|QBasic|1.1}} |
{{works with|QBasic|1.1}} |
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{{works with|QuickBasic|4.5}} |
{{works with|QuickBasic|4.5}} |
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< |
<syntaxhighlight lang="qbasic">h = 0! |
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PRINT "The first twenty harmonic numbers are:" |
PRINT "The first twenty harmonic numbers are:" |
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PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1 |
PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1 |
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NEXT i |
NEXT i |
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END</ |
END</syntaxhighlight> |
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==={{header|True BASIC}}=== |
==={{header|True BASIC}}=== |
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< |
<syntaxhighlight lang="qbasic">LET h = 0 |
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PRINT "The first twenty harmonic numbers are:" |
PRINT "The first twenty harmonic numbers are:" |
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PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1 |
PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1 |
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NEXT i |
NEXT i |
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END</ |
END</syntaxhighlight> |
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==={{header|Yabasic}}=== |
==={{header|Yabasic}}=== |
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< |
<syntaxhighlight lang="freebasic">h = 0.0 |
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print "The first twenty harmonic numbers are:" |
print "The first twenty harmonic numbers are:" |
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print "The first harmonic number greater than ", i, " is ", h, ", at position ", n-1 |
print "The first harmonic number greater than ", i, " is ", h, ", at position ", n-1 |
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next i |
next i |
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end</ |
end</syntaxhighlight> |
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=={{header|C++}}== |
=={{header|C++}}== |
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{{libheader|Boost}} |
{{libheader|Boost}} |
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< |
<syntaxhighlight lang="cpp">#include <iomanip> |
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#include <iostream> |
#include <iostream> |
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#include <boost/rational.hpp> |
#include <boost/rational.hpp> |
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std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n'; |
std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n'; |
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} |
} |
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}</ |
}</syntaxhighlight> |
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{{out}} |
{{out}} |
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=={{header|COBOL}}== |
=={{header|COBOL}}== |
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< |
<syntaxhighlight lang="cobol"> IDENTIFICATION DIVISION. |
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PROGRAM-ID. HARMONIC. |
PROGRAM-ID. HARMONIC. |
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NEXT-HARMONIC. |
NEXT-HARMONIC. |
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ADD 1 TO N. |
ADD 1 TO N. |
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COMPUTE HN = HN + 1 / N.</ |
COMPUTE HN = HN + 1 / N.</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre>First 20 harmonic numbers: |
<pre>First 20 harmonic numbers: |
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{{works with|Factor|0.99 2021-02-05}} |
{{works with|Factor|0.99 2021-02-05}} |
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< |
<syntaxhighlight lang="factor">USING: formatting grouping io kernel lists lists.lazy math |
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math.functions math.ranges math.statistics math.text.english |
math.functions math.ranges math.statistics math.text.english |
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prettyprint sequences tools.memory.private ; |
prettyprint sequences tools.memory.private ; |
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dup first-gt [ commas ] [ 1 + number>text ] bi |
dup first-gt [ commas ] [ 1 + number>text ] bi |
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" greater than %2d: %6s (term number %s)\n" printf |
" greater than %2d: %6s (term number %s)\n" printf |
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] each</ |
] each</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=={{header|Forth}}== |
=={{header|Forth}}== |
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Uses fixed point computation which is more traditional in FORTH. |
Uses fixed point computation which is more traditional in FORTH. |
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<syntaxhighlight lang="forth"> |
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<lang Forth> |
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warnings off |
warnings off |
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." The nth index of the first harmonic number that exceeds the nth integer: " cr task2 cr |
." The nth index of the first harmonic number that exceeds the nth integer: " cr task2 cr |
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bye |
bye |
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</syntaxhighlight> |
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</lang> |
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{{Out}} |
{{Out}} |
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<pre> |
<pre> |
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=={{header|FreeBASIC}}== |
=={{header|FreeBASIC}}== |
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< |
<syntaxhighlight lang="freebasic">dim as double h = 0.0 |
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dim as uinteger n, i |
dim as uinteger n, i |
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wend |
wend |
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print "The first harmonic number greater than ";i;" is ";h;", at position ";n-1 |
print "The first harmonic number greater than ";i;" is ";h;", at position ";n-1 |
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next i</ |
next i</syntaxhighlight> |
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{{out}}<pre>The first twenty harmonic numbers are: |
{{out}}<pre>The first twenty harmonic numbers are: |
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1 1 |
1 1 |
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=={{header|Go}}== |
=={{header|Go}}== |
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{{trans|Wren}} |
{{trans|Wren}} |
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< |
<syntaxhighlight lang="go">package main |
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import ( |
import ( |
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} |
} |
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} |
} |
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}</ |
}</syntaxhighlight> |
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{{out}} |
{{out}} |
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=={{header|Haskell}}== |
=={{header|Haskell}}== |
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< |
<syntaxhighlight lang="haskell">import Data.List (find) |
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import Data.Ratio |
import Data.Ratio |
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showRatio = |
showRatio = |
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((<>) . show . numerator) |
((<>) . show . numerator) |
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<*> (('/' :) . show . denominator)</ |
<*> (('/' :) . show . denominator)</syntaxhighlight> |
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{{Out}} |
{{Out}} |
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This entry requires a rational arithmetic module such as is available |
This entry requires a rational arithmetic module such as is available |
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at [[Arithmetic/Rational#jq]]. |
at [[Arithmetic/Rational#jq]]. |
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< |
<syntaxhighlight lang="jq"># include "rational"; # a reminder |
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def harmonic: |
def harmonic: |
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select(.emit).emit) ); |
select(.emit).emit) ); |
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task1, "", task2(10)</ |
task1, "", task2(10)</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=={{header|Julia}}== |
=={{header|Julia}}== |
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< |
<syntaxhighlight lang="julia">const memoizer = [BigFloat(1.0), BigFloat(1.5)] |
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""" |
""" |
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testharmonics() |
testharmonics() |
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</ |
</syntaxhighlight>{{out}} |
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<pre> |
<pre> |
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harmonic(1) = 1.0 |
harmonic(1) = 1.0 |
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=== Using rationals === |
=== Using rationals === |
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< |
<syntaxhighlight lang="julia">const harmonics = accumulate((x, y) -> x + big"1" // y, 1:12370) |
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println("First twenty harmonic numbers as rationals:") |
println("First twenty harmonic numbers as rationals:") |
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print("First Harmonic > $n is at position $idx and is: ", harmonics[idx], "\n\n") |
print("First Harmonic > $n is at position $idx and is: ", harmonics[idx], "\n\n") |
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end |
end |
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</ |
</syntaxhighlight>{{out}} |
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<pre> |
<pre> |
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First twenty harmonic numbers as rationals: |
First twenty harmonic numbers as rationals: |
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=={{header|Mathematica}}/{{header|Wolfram Language}}== |
=={{header|Mathematica}}/{{header|Wolfram Language}}== |
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< |
<syntaxhighlight lang="mathematica">nums = HarmonicNumber[Range[15000]]; |
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nums[[;; 20]] |
nums[[;; 20]] |
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LengthWhile[nums, LessEqualThan[#]] + 1 & /@ Range[10]</ |
LengthWhile[nums, LessEqualThan[#]] + 1 & /@ Range[10]</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre>{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504} |
<pre>{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504} |
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=={{header|Nim}}== |
=={{header|Nim}}== |
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===Using floats=== |
===Using floats=== |
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< |
<syntaxhighlight lang="nim">import strformat |
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iterator h(): (int, float) = |
iterator h(): (int, float) = |
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echo &"Index of the first term greater than {target.int:2}: {idx}" |
echo &"Index of the first term greater than {target.int:2}: {idx}" |
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if target == 10: break |
if target == 10: break |
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else: target += 1</ |
else: target += 1</syntaxhighlight> |
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{{out}} |
{{out}} |
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===Using big integers=== |
===Using big integers=== |
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{{libheader|bignum}} |
{{libheader|bignum}} |
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< |
<syntaxhighlight lang="nim">import strformat |
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import bignum |
import bignum |
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echo &"Index of the first term greater than {target:2}: {idx}" |
echo &"Index of the first term greater than {target:2}: {idx}" |
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if target == 10: break |
if target == 10: break |
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else: inc target</ |
else: inc target</syntaxhighlight> |
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{{out}} |
{{out}} |
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=={{header|PARI/GP}}== |
=={{header|PARI/GP}}== |
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< |
<syntaxhighlight lang="parigp">h=0 |
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for(n=1,20,h=h+1/n;print(n," ",h)) |
for(n=1,20,h=h+1/n;print(n," ",h)) |
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h=0; n=1 |
h=0; n=1 |
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for(i=1,10,while(h<i,h=h+1/n;n=n+1);print(n-1))</ |
for(i=1,10,while(h<i,h=h+1/n;n=n+1);print(n-1))</syntaxhighlight> |
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=={{header|Perl}}== |
=={{header|Perl}}== |
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< |
<syntaxhighlight lang="perl">use strict; |
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use warnings; |
use warnings; |
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use feature 'say'; |
use feature 'say'; |
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for my $i (1..10) { |
for my $i (1..10) { |
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printf " greater than %2d: %5s\n", $i, firstidx { $_ > $i } @H; |
printf " greater than %2d: %5s\n", $i, firstidx { $_ > $i } @H; |
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}</ |
}</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre>First twenty harmonic numbers as rationals: |
<pre>First twenty harmonic numbers as rationals: |
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=={{header|Phix}}== |
=={{header|Phix}}== |
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{{libheader|Phix/mpfr}} |
{{libheader|Phix/mpfr}} |
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<!--< |
<!--<syntaxhighlight lang="phix">(phixonline)--> |
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<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"0.8.4"</span><span style="color: #0000FF;">)</span> |
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"0.8.4"</span><span style="color: #0000FF;">)</span> |
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<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> |
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> |
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<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" greater than %2d: %,6d (%s term)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">gt</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #7060A8;">ordinal</span><span style="color: #0000FF;">(</span><span style="color: #000000;">gt</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])})</span> |
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" greater than %2d: %,6d (%s term)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">gt</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span><span style="color: #7060A8;">ordinal</span><span style="color: #0000FF;">(</span><span style="color: #000000;">gt</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])})</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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<!--</ |
<!--</syntaxhighlight>--> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=== using standard floats === |
=== using standard floats === |
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<!--< |
<!--<syntaxhighlight lang="phix">(phixonline)--> |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">gn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
<span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">gn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
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<span style="color: #004080;">atom</span> <span style="color: #000000;">hn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
<span style="color: #004080;">atom</span> <span style="color: #000000;">hn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span> |
<span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span> |
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<!--</ |
<!--</syntaxhighlight>--> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |
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=={{header|Prolog}}== |
=={{header|Prolog}}== |
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{{works with|SWI Prolog}} |
{{works with|SWI Prolog}} |
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< |
<syntaxhighlight lang="prolog">main:- |
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print_harmonic_series(20), |
print_harmonic_series(20), |
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nl, |
nl, |
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nth_harmonic_number(N, T, H1):- |
nth_harmonic_number(N, T, H1):- |
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harmonic_next(H1, H2), |
harmonic_next(H1, H2), |
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nth_harmonic_number(N, T, H2).</ |
nth_harmonic_number(N, T, H2).</syntaxhighlight> |
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{{out}} |
{{out}} |
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=={{header|Python}}== |
=={{header|Python}}== |
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A generator function using fractions: |
A generator function using fractions: |
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< |
<syntaxhighlight lang="python">from fractions import Fraction |
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def harmonic_series(): |
def harmonic_series(): |
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from itertools import islice |
from itertools import islice |
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for n, d in (h.as_integer_ratio() for h in islice(harmonic_series(), 20)): |
for n, d in (h.as_integer_ratio() for h in islice(harmonic_series(), 20)): |
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print(n, '/', d)</ |
print(n, '/', d)</syntaxhighlight> |
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{{out}} |
{{out}} |
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Or alternatively, in terms of itertools.accumulate: |
Or alternatively, in terms of itertools.accumulate: |
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< |
<syntaxhighlight lang="python">'''Harmonic series''' |
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from fractions import Fraction |
from fractions import Fraction |
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# MAIN --- |
# MAIN --- |
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if __name__ == '__main__': |
if __name__ == '__main__': |
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main()</ |
main()</syntaxhighlight> |
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{{Out}} |
{{Out}} |
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<pre>First 20 terms of the harmonic series: |
<pre>First 20 terms of the harmonic series: |
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=={{header|Quackery}}== |
=={{header|Quackery}}== |
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< |
<syntaxhighlight lang="quackery"> [ $ "bigrat.qky" loadfile ] now! |
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0 n->v |
0 n->v |
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Line 1,527: | Line 1,527: | ||
again ] |
again ] |
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temp release |
temp release |
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drop 2drop</ |
drop 2drop</syntaxhighlight> |
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{{out}} |
{{out}} |
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===Direct Summation=== |
===Direct Summation=== |
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The talk page helpfully points out that we can be remarkably lazy here. |
The talk page helpfully points out that we can be remarkably lazy here. |
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< |
<syntaxhighlight lang="rsplus">HofN <- function(n) sum(1/seq_len(n)) #Task 1 |
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H <- sapply(1:100000, HofN) |
H <- sapply(1:100000, HofN) |
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print(H[1:20]) #Task 2 |
print(H[1:20]) #Task 2 |
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print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</ |
print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</syntaxhighlight> |
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{{out}} |
{{out}} |
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<pre>> print(H[1:20]) #Task 2 |
<pre>> print(H[1:20]) #Task 2 |
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===Cumulative Sums=== |
===Cumulative Sums=== |
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As for doing this properly, R provides a handy cumsum function. |
As for doing this properly, R provides a handy cumsum function. |
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< |
<syntaxhighlight lang="rsplus">firstNHarmonicNumbers <- function(n) cumsum(1/seq_len(n)) #Task 1 |
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H <- firstNHarmonicNumbers(100000) #Runs stunningly quick |
H <- firstNHarmonicNumbers(100000) #Runs stunningly quick |
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print(H[1:20]) #Task 2 |
print(H[1:20]) #Task 2 |
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print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</ |
print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch</syntaxhighlight> |
||
=={{header|Raku}}== |
=={{header|Raku}}== |
||
Using [https://modules.raku.org/search/?q=Lingua%3A%3AEN%3A%3ANumbers Lingua::EN::Numbers] from the [https://modules.raku.org/ Raku ecosystem]. |
Using [https://modules.raku.org/search/?q=Lingua%3A%3AEN%3A%3ANumbers Lingua::EN::Numbers] from the [https://modules.raku.org/ Raku ecosystem]. |
||
<lang |
<syntaxhighlight lang="raku" line>use Lingua::EN::Numbers; |
||
my @H = [\+] (1..*).map: { FatRat.new: 1, $_ }; |
my @H = [\+] (1..*).map: { FatRat.new: 1, $_ }; |
||
Line 1,598: | Line 1,598: | ||
say "\n(zero based) Index of first value:"; |
say "\n(zero based) Index of first value:"; |
||
printf " greater than %2d: %6s (%s term)\n", |
printf " greater than %2d: %6s (%s term)\n", |
||
$_, comma( my $i = @H.first(* > $_, :k) ), ordinal 1 + $i for 1..10;</ |
$_, comma( my $i = @H.first(* > $_, :k) ), ordinal 1 + $i for 1..10;</syntaxhighlight> |
||
{{out}} |
{{out}} |
||
<pre>First twenty harmonic numbers as rationals: |
<pre>First twenty harmonic numbers as rationals: |
||
Line 1,623: | Line 1,623: | ||
=={{header|REXX}}== |
=={{header|REXX}}== |
||
The default number of decimal digits (9) could've been used instead of '''80''' for this task's particular limits. |
The default number of decimal digits (9) could've been used instead of '''80''' for this task's particular limits. |
||
< |
<syntaxhighlight lang="rexx">/*REXX pgm to calculate N numbers (sums) in the harmonic series and also when they > X. */ |
||
parse arg digs sums high ints /*obtain optional arguments from the CL*/ |
parse arg digs sums high ints /*obtain optional arguments from the CL*/ |
||
if digs='' | digs="," then digs= 80 /*Not specified? Then use the default.*/ |
if digs='' | digs="," then digs= 80 /*Not specified? Then use the default.*/ |
||
Line 1,656: | Line 1,656: | ||
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
/*──────────────────────────────────────────────────────────────────────────────────────*/ |
||
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? |
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? |
||
th: parse arg x; return word('th st nd rd', 1 + (x//10) *(x//100%10\==1) *(x//10<4))</ |
th: parse arg x; return word('th st nd rd', 1 + (x//10) *(x//100%10\==1) *(x//10<4))</syntaxhighlight> |
||
{{out|output|text= when using the default inputs:}} |
{{out|output|text= when using the default inputs:}} |
||
<pre> |
<pre> |
||
Line 1,693: | Line 1,693: | ||
=={{header|Ring}}== |
=={{header|Ring}}== |
||
< |
<syntaxhighlight lang="ring"> |
||
decimals(12) |
decimals(12) |
||
sum = 0 |
sum = 0 |
||
Line 1,722: | Line 1,722: | ||
next |
next |
||
next |
next |
||
</syntaxhighlight> |
|||
</lang> |
|||
{{out}} |
{{out}} |
||
<pre> |
<pre> |
||
Line 1,761: | Line 1,761: | ||
=={{header|Rust}}== |
=={{header|Rust}}== |
||
Using big rationals and big integers from the [https://docs.rs/num/latest/num/ num] crate. |
Using big rationals and big integers from the [https://docs.rs/num/latest/num/ num] crate. |
||
< |
<syntaxhighlight lang="rust"> |
||
use num::rational::Ratio; |
use num::rational::Ratio; |
||
use num::{BigInt, FromPrimitive}; |
use num::{BigInt, FromPrimitive}; |
||
Line 1,797: | Line 1,797: | ||
result |
result |
||
} |
} |
||
</syntaxhighlight> |
|||
</lang> |
|||
{{out}} |
{{out}} |
||
Line 1,831: | Line 1,831: | ||
=={{header|Verilog}}== |
=={{header|Verilog}}== |
||
< |
<syntaxhighlight lang="verilog">module main; |
||
integer n, i; |
integer n, i; |
||
real h; |
real h; |
||
Line 1,857: | Line 1,857: | ||
$finish ; |
$finish ; |
||
end |
end |
||
endmodule</ |
endmodule</syntaxhighlight> |
||
=={{header|Wren}}== |
=={{header|Wren}}== |
||
{{libheader|Wren-big}} |
{{libheader|Wren-big}} |
||
{{libheader|Wren-fmt}} |
{{libheader|Wren-fmt}} |
||
< |
<syntaxhighlight lang="ecmascript">import "/big" for BigRat |
||
import "/fmt" for Fmt |
import "/fmt" for Fmt |
||
Line 1,886: | Line 1,886: | ||
} |
} |
||
n = n + 1 |
n = n + 1 |
||
}</ |
}</syntaxhighlight> |
||
{{out}} |
{{out}} |
||
Line 1,927: | Line 1,927: | ||
=={{header|XPL0}}== |
=={{header|XPL0}}== |
||
< |
<syntaxhighlight lang="xpl0">func real Harmonic(N); \Return Nth harmonic number |
||
int N; real X; |
int N; real X; |
||
[X:= 1.0; |
[X:= 1.0; |
||
Line 1,948: | Line 1,948: | ||
CrLf(0); |
CrLf(0); |
||
]; |
]; |
||
]</ |
]</syntaxhighlight> |
||
{{out}} |
{{out}} |
Revision as of 14:52, 27 August 2022
You are encouraged to solve this task according to the task description, using any language you may know.
This page uses content from Wikipedia. The original article was at Harmonic number. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
Hn = 1 + 1/2 + 1/3 + ... + 1/n
The series of harmonic numbers thus obtained is often loosely referred to as the harmonic series.
Harmonic numbers are closely related to the Riemann zeta function, and roughly approximate the natural logarithm function; differing by γ (lowercase Gamma), the Euler–Mascheroni constant.
The harmonic series is divergent, albeit quite slowly, and grows toward infinity.
- Task
- Write a function (routine, procedure, whatever it may be called in your language) to generate harmonic numbers.
- Use that procedure to show the values of the first 20 harmonic numbers.
- Find and show the position in the series of the first value greater than the integers 1 through 5
- Stretch
- Find and show the position in the series of the first value greater than the integers 6 through 10
- Related
ALGOL 68
Using standard lenghth REAL numbers this can find the first Harmonic number > 10, would probably need higher precision to find Harmonic numbers with larger values.
BEGIN # find some harmonic numbers, Hn is the sum if the reciprocals of 1..n #
# returns the first n Harmonic numbers #
OP HARMONIC = ( INT n )[]REAL:
BEGIN
[ 1 : n ]REAL h;
h[ 1 ] := 1;
FOR i FROM 2 TO n DO
h[ i ] := h[ i - 1 ] + ( 1 / i )
OD;
h
END # HARMONIC # ;
# find the first 20 000 harmonic numbers #
[]REAL h = HARMONIC 20 000;
# show the first 20 harmonic numbers #
FOR i TO 20 DO
print( ( whole( i, -2 ), ":", fixed( h[ i ], -14, 8 ), newline ) )
OD;
# find the positions of the first harmonic number > n where n in 1... #
INT rqd int := 1;
REAL rqd real := 1;
FOR i TO UPB h DO
IF h[ i ] > rqd real THEN
# found the first harmonic number greater than rqd real #
print( ( "Position of the first harmonic number > ", whole( rqd int, -2 ), ": ", whole( i, 0 ), newline ) );
rqd int +:= 1;
rqd real +:= 1
FI
OD
END
- Output:
1: 1.00000000 2: 1.50000000 3: 1.83333333 4: 2.08333333 5: 2.28333333 6: 2.45000000 7: 2.59285714 8: 2.71785714 9: 2.82896825 10: 2.92896825 11: 3.01987734 12: 3.10321068 13: 3.18013376 14: 3.25156233 15: 3.31822899 16: 3.38072899 17: 3.43955252 18: 3.49510808 19: 3.54773966 20: 3.59773966 Position of the first harmonic number > 1: 2 Position of the first harmonic number > 2: 4 Position of the first harmonic number > 3: 11 Position of the first harmonic number > 4: 31 Position of the first harmonic number > 5: 83 Position of the first harmonic number > 6: 227 Position of the first harmonic number > 7: 616 Position of the first harmonic number > 8: 1674 Position of the first harmonic number > 9: 4550 Position of the first harmonic number > 10: 12367
Arturo
H: function [n][
sum map 1..n => reciprocal
]
firstAbove: function [lim][
i: 1
while ø [
if lim < to :floating H i ->
return i
i: i + 1
]
]
print "The first 20 harmonic numbers:"
print map 1..20 => H
print ""
loop 1..4 'l [
print ["Position of first term >" l ":" firstAbove l]
]
- Output:
The first 20 harmonic numbers: 1/1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 Position of first term > 1 : 2 Position of first term > 2 : 4 Position of first term > 3 : 11 Position of first term > 4 : 31
AWK
# syntax: GAWK -f HARMONIC_SERIES.AWK
# converted from FreeBASIC
BEGIN {
limit = 20
printf("The first %d harmonic numbers:\n",limit)
for (n=1; n<=limit; n++) {
h += 1/n
printf("%2d %11.8f\n",n,h)
}
print("")
h = 1
n = 2
for (i=2; i<=10; i++) {
while (h < i) {
h += 1/n
n++
}
printf("The first harmonic number > %2d is %11.8f at position %d\n",i,h,n-1)
}
exit(0)
}
- Output:
The first 20 harmonic numbers: 1 1.00000000 2 1.50000000 3 1.83333333 4 2.08333333 5 2.28333333 6 2.45000000 7 2.59285714 8 2.71785714 9 2.82896825 10 2.92896825 11 3.01987734 12 3.10321068 13 3.18013376 14 3.25156233 15 3.31822899 16 3.38072899 17 3.43955252 18 3.49510808 19 3.54773966 20 3.59773966 The first harmonic number > 2 is 2.08333333 at position 4 The first harmonic number > 3 is 3.01987734 at position 11 The first harmonic number > 4 is 4.02724520 at position 31 The first harmonic number > 5 is 5.00206827 at position 83 The first harmonic number > 6 is 6.00436671 at position 227 The first harmonic number > 7 is 7.00127410 at position 616 The first harmonic number > 8 is 8.00048557 at position 1674 The first harmonic number > 9 is 9.00020806 at position 4550 The first harmonic number > 10 is 10.00004301 at position 12367
BASIC
BASIC256
h = 0.0
print "The first twenty harmonic numbers are:"
for n = 1 to 20
h += 1.0 / n
print n, h
next n
print
h = 1 : n = 2
for i = 2 to 10
while h < i
h += 1.0 / n
n += 1
end while
print "The first harmonic number greater than "; i; " is "; h; ", at position "; n-1
next i
end
QBasic
h = 0!
PRINT "The first twenty harmonic numbers are:"
FOR n = 1 TO 20
h = h + 1! / n
PRINT n, h
NEXT n
PRINT
h = 1: n = 2
FOR i = 2 TO 10
WHILE h < i
h = h + 1! / n
n = n + 1
WEND
PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1
NEXT i
END
True BASIC
LET h = 0
PRINT "The first twenty harmonic numbers are:"
FOR n = 1 TO 20
LET h = h + 1 / n
PRINT n, h
NEXT n
PRINT
LET h = 1
LET n = 2
FOR i = 2 TO 10
DO WHILE h < i
LET h = h + 1 / n
LET n = n + 1
LOOP
PRINT "The first harmonic number greater than "; i; " is "; h; ", at position "; n - 1
NEXT i
END
Yabasic
h = 0.0
print "The first twenty harmonic numbers are:"
for n = 1 to 20
h = h + 1.0 / n
print n, chr$(9), h
next n
print
h = 1 : n = 2
for i = 2 to 10
while h < i
h = h + 1.0 / n
n = n + 1
wend
print "The first harmonic number greater than ", i, " is ", h, ", at position ", n-1
next i
end
C++
#include <iomanip>
#include <iostream>
#include <boost/rational.hpp>
#include <boost/multiprecision/gmp.hpp>
using integer = boost::multiprecision::mpz_int;
using rational = boost::rational<integer>;
class harmonic_generator {
public:
rational next() {
rational result = term_;
term_ += rational(1, ++n_);
return result;
}
void reset() {
n_ = 1;
term_ = 1;
}
private:
integer n_ = 1;
rational term_ = 1;
};
int main() {
std::cout << "First 20 harmonic numbers:\n";
harmonic_generator hgen;
for (int i = 1; i <= 20; ++i)
std::cout << std::setw(2) << i << ". " << hgen.next() << '\n';
rational h;
for (int i = 1; i <= 80; ++i)
h = hgen.next();
std::cout << "\n100th harmonic number: " << h << "\n\n";
int n = 1;
hgen.reset();
for (int i = 1; n <= 10; ++i) {
if (hgen.next() > n)
std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n';
}
}
- Output:
First 20 harmonic numbers: 1. 1/1 2. 3/2 3. 11/6 4. 25/12 5. 137/60 6. 49/20 7. 363/140 8. 761/280 9. 7129/2520 10. 7381/2520 11. 83711/27720 12. 86021/27720 13. 1145993/360360 14. 1171733/360360 15. 1195757/360360 16. 2436559/720720 17. 42142223/12252240 18. 14274301/4084080 19. 275295799/77597520 20. 55835135/15519504 100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 Position of first term > 1: 2 Position of first term > 2: 4 Position of first term > 3: 11 Position of first term > 4: 31 Position of first term > 5: 83 Position of first term > 6: 227 Position of first term > 7: 616 Position of first term > 8: 1674 Position of first term > 9: 4550 Position of first term > 10: 12367
COBOL
IDENTIFICATION DIVISION.
PROGRAM-ID. HARMONIC.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 VARS.
03 N PIC 9(5) VALUE ZERO.
03 HN PIC 9(2)V9(12) VALUE ZERO.
03 INT PIC 99 VALUE ZERO.
01 OUT-VARS.
03 POS PIC Z(4)9.
03 FILLER PIC X(3) VALUE SPACES.
03 H-OUT PIC Z9.9(12).
PROCEDURE DIVISION.
BEGIN.
DISPLAY "First 20 harmonic numbers:"
PERFORM SHOW-HARMONIC 20 TIMES.
DISPLAY SPACES.
MOVE ZERO TO N, HN.
DISPLAY "First harmonic number to exceed whole number:"
PERFORM EXCEED-INT 10 TIMES.
STOP RUN.
SHOW-HARMONIC.
PERFORM NEXT-HARMONIC.
MOVE HN TO H-OUT.
DISPLAY H-OUT.
EXCEED-INT.
ADD 1 TO INT.
PERFORM NEXT-HARMONIC UNTIL HN IS GREATER THAN INT.
MOVE N TO POS.
MOVE HN TO H-OUT.
DISPLAY OUT-VARS.
NEXT-HARMONIC.
ADD 1 TO N.
COMPUTE HN = HN + 1 / N.
- Output:
First 20 harmonic numbers: 1.000000000000 1.500000000000 1.833333333333 2.083333333333 2.283333333333 2.449999999999 2.592857142856 2.717857142856 2.828968253967 2.928968253967 3.019877344876 3.103210678209 3.180133755132 3.251562326560 3.318228993226 3.380728993226 3.439552522637 3.495108078192 3.547739657139 3.597739657139 First harmonic number to exceed whole number: 2 1.500000000000 4 2.083333333333 11 3.019877344876 31 4.027245195428 83 5.002068272651 227 6.004366708257 616 7.001274096877 1674 8.000485571261 4550 9.000208060802 12367 10.000043002313
Factor
This solution uses the following (rather accurate) approximation of the harmonic numbers to find the first indices greater than the integers:
Hn ≈ ln(n) + γ + 1/2n - 1/12n2
where γ
is the Euler-Mascheroni constant, approximately 0.5772156649
.
USING: formatting grouping io kernel lists lists.lazy math
math.functions math.ranges math.statistics math.text.english
prettyprint sequences tools.memory.private ;
! Euler-Mascheroni constant
CONSTANT: γ 0.5772156649
: Hn-approx ( n -- ~Hn )
[ log γ + 1 2 ] [ * /f + 1 ] [ sq 12 * /f - ] tri ;
: lharmonics ( -- list ) 1 lfrom [ Hn-approx ] lmap-lazy ;
: first-gt ( m -- n ) lharmonics swap '[ _ < ] lwhile llength ;
"First twenty harmonic numbers as mixed numbers:" print
100 [1,b] [ recip ] map cum-sum
[ 20 head 5 group simple-table. nl ]
[ "One hundredth:" print last . nl ] bi
"(zero based) Index of first value:" print
10 [1,b] [
dup first-gt [ commas ] [ 1 + number>text ] bi
" greater than %2d: %6s (term number %s)\n" printf
] each
- Output:
First twenty harmonic numbers as mixed numbers: 1 1+1/2 1+5/6 2+1/12 2+17/60 2+9/20 2+83/140 2+201/280 2+2089/2520 2+2341/2520 3+551/27720 3+2861/27720 3+64913/360360 3+90653/360360 3+114677/360360 3+274399/720720 3+5385503/12252240 3+2022061/4084080 3+42503239/77597520 3+9276623/15519504 One hundredth: 5+522561233577855727314756256041670736351/2788815009188499086581352357412492142272 (zero based) Index of first value: greater than 1: 1 (term number two) greater than 2: 3 (term number four) greater than 3: 10 (term number eleven) greater than 4: 30 (term number thirty-one) greater than 5: 82 (term number eighty-three) greater than 6: 226 (term number two hundred and twenty-seven) greater than 7: 615 (term number six hundred and sixteen) greater than 8: 1,673 (term number one thousand, six hundred and seventy-four) greater than 9: 4,549 (term number four thousand, five hundred and fifty) greater than 10: 12,366 (term number twelve thousand, three hundred and sixty-seven)
Forth
Uses fixed point computation which is more traditional in FORTH.
warnings off
1.000.000.000.000.000 drop constant 1.0fx \ fractional part is 15 decimal digits.
: .h ( n -- )
s>d <# 14 for # next [char] . hold #s #> type space ;
1.0fx 1 2constant first-harmonic
: round 5 + 10 / ;
: next-harmonic ( h n -- h' n' )
1+ tuck [ 1.0fx 10 * ] literal swap / round + swap ;
: task1
first-harmonic 19 for over cr .h next-harmonic next 2drop ;
: task2
first-harmonic
11 1 do
begin over i 1.0fx * <= while
next-harmonic
repeat
dup .
loop 2drop ;
." The first 10 harmonic numbers: " task1 cr cr
." The nth index of the first harmonic number that exceeds the nth integer: " cr task2 cr
bye
- Output:
The first 10 harmonic numbers: 1.000000000000000 1.500000000000000 1.833333333333333 2.083333333333333 2.283333333333333 2.450000000000000 2.592857142857143 2.717857142857143 2.828968253968254 2.928968253968254 3.019877344877345 3.103210678210678 3.180133755133755 3.251562326562326 3.318228993228993 3.380728993228993 3.439552522640758 3.495108078196314 3.547739657143682 3.597739657143682 The nth index of the first harmonic number that exceeds the nth integer: 2 4 11 31 83 227 616 1674 4550 12367
FreeBASIC
dim as double h = 0.0
dim as uinteger n, i
print "The first twenty harmonic numbers are:"
for n = 1 to 20
h += 1.0/n
print n, h
next n
h = 1 : n = 2
for i=2 to 10
while h<i
h+=1.0/n
n+=1
wend
print "The first harmonic number greater than ";i;" is ";h;", at position ";n-1
next i
- Output:
The first twenty harmonic numbers are:1 1 2 1.5 3 1.833333333333333 4 2.083333333333333 5 2.283333333333333 6 2.45 7 2.592857142857143 8 2.717857142857143 9 2.828968253968254 10 2.928968253968254 11 3.019877344877345 12 3.103210678210678 13 3.180133755133755 14 3.251562326562327 15 3.318228993228994 16 3.380728993228994 17 3.439552522640758 18 3.495108078196314 19 3.547739657143682 20 3.597739657143682 The first harmonic number greater than 2 is 2.083333333333333, at position 4 The first harmonic number greater than 3 is 3.019877344877345, at position 11 The first harmonic number greater than 4 is 4.02724519543652, at position 31 The first harmonic number greater than 5 is 5.002068272680166, at position 83 The first harmonic number greater than 6 is 6.004366708345567, at position 227 The first harmonic number greater than 7 is 7.001274097134162, at position 616 The first harmonic number greater than 8 is 8.000485571995782, at position 1674 The first harmonic number greater than 9 is 9.000208062931115, at position 4550
The first harmonic number greater than 10 is 10.00004300827578, at position 12367
Go
package main
import (
"fmt"
"math/big"
)
func harmonic(n int) *big.Rat {
sum := new(big.Rat)
for i := int64(1); i <= int64(n); i++ {
r := big.NewRat(1, i)
sum.Add(sum, r)
}
return sum
}
func main() {
fmt.Println("The first 20 harmonic numbers and the 100th, expressed in rational form, are:")
numbers := make([]int, 21)
for i := 1; i <= 20; i++ {
numbers[i-1] = i
}
numbers[20] = 100
for _, i := range numbers {
fmt.Printf("%3d : %s\n", i, harmonic(i))
}
fmt.Println("\nThe first harmonic number to exceed the following integers is:")
const limit = 10
for i, n, h := 1, 1, 0.0; i <= limit; n++ {
h += 1.0 / float64(n)
if h > float64(i) {
fmt.Printf("integer = %2d -> n = %6d -> harmonic number = %9.6f (to 6dp)\n", i, n, h)
i++
}
}
}
- Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are: 1 : 1/1 2 : 3/2 3 : 11/6 4 : 25/12 5 : 137/60 6 : 49/20 7 : 363/140 8 : 761/280 9 : 7129/2520 10 : 7381/2520 11 : 83711/27720 12 : 86021/27720 13 : 1145993/360360 14 : 1171733/360360 15 : 1195757/360360 16 : 2436559/720720 17 : 42142223/12252240 18 : 14274301/4084080 19 : 275295799/77597520 20 : 55835135/15519504 100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 The first harmonic number to exceed the following integers is: integer = 1 -> n = 2 -> harmonic number = 1.500000 (to 6dp) integer = 2 -> n = 4 -> harmonic number = 2.083333 (to 6dp) integer = 3 -> n = 11 -> harmonic number = 3.019877 (to 6dp) integer = 4 -> n = 31 -> harmonic number = 4.027245 (to 6dp) integer = 5 -> n = 83 -> harmonic number = 5.002068 (to 6dp) integer = 6 -> n = 227 -> harmonic number = 6.004367 (to 6dp) integer = 7 -> n = 616 -> harmonic number = 7.001274 (to 6dp) integer = 8 -> n = 1674 -> harmonic number = 8.000486 (to 6dp) integer = 9 -> n = 4550 -> harmonic number = 9.000208 (to 6dp) integer = 10 -> n = 12367 -> harmonic number = 10.000043 (to 6dp)
Haskell
import Data.List (find)
import Data.Ratio
--------------------- HARMONIC SERIES --------------------
harmonic :: [Rational]
harmonic =
scanl1
(\a x -> a + 1 / x)
[1 ..]
-------------------------- TESTS -------------------------
main :: IO ()
main = do
putStrLn "First 20 terms:"
mapM_ putStrLn $
showRatio <$> take 20 harmonic
putStrLn "\n100th term:"
putStrLn $ showRatio (harmonic !! 99)
putStrLn ""
putStrLn "One-based indices of first terms above threshold values:"
let indexedHarmonic = zip [0 ..] harmonic
mapM_
putStrLn
$ fmap
( showFirstLimit
<*> \n -> find ((> n) . snd) indexedHarmonic
)
[1 .. 10]
-------------------- DISPLAY FORMATTING ------------------
showFirstLimit n (Just (i, r)) =
"Term "
<> show (succ i)
<> " is the first above "
<> show (numerator n)
showRatio :: Ratio Integer -> String
showRatio =
((<>) . show . numerator)
<*> (('/' :) . show . denominator)
- Output:
First 20 terms: 1/1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 100th term: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 One-based indices of first terms above threshold values: Term 2 is the first above 1 Term 4 is the first above 2 Term 11 is the first above 3 Term 31 is the first above 4 Term 83 is the first above 5 Term 227 is the first above 6 Term 616 is the first above 7 Term 1674 is the first above 8 Term 4550 is the first above 9 Term 12367 is the first above 10
jq
Works with gojq, the Go implementation of jq
This entry requires a rational arithmetic module such as is available at Arithmetic/Rational#jq.
# include "rational"; # a reminder
def harmonic:
reduce range(1; 1+.) as $i ( r(0;1);
radd(.; r(1; $i) ));
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def task1:
"The first 20 harmonic numbers and the 100th, expressed in rational form, are:",
(range(1;21), 100
| "\(.) : \(harmonic|rpp)" );
def task2($limit):
"The first harmonic number to exceed the following integers is:",
limit($limit;
foreach range(0; infinite) as $n (
{i: 1, n: 1, h: r(0;1)};
.emit = false
| .h = radd(.h; r(1; .n))
| .i as $i
| if .h | rgreaterthan($i)
then .emit = "integer = \(.i|lpad(2)) -> n = \(.n| lpad(6)) -> harmonic number = \(.h|r_to_decimal(6)) (to 6dp)"
| .i += 1
else .
end
| .n += 1;
select(.emit).emit) );
task1, "", task2(10)
- Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are: 1 : 1 // 1 2 : 3 // 2 3 : 11 // 6 4 : 25 // 12 5 : 137 // 60 6 : 49 // 20 7 : 363 // 140 8 : 761 // 280 9 : 7129 // 2520 10 : 7381 // 2520 11 : 83711 // 27720 12 : 86021 // 27720 13 : 1145993 // 360360 14 : 1171733 // 360360 15 : 1195757 // 360360 16 : 2436559 // 720720 17 : 42142223 // 12252240 18 : 14274301 // 4084080 19 : 275295799 // 77597520 20 : 55835135 // 15519504 100 : 14466636279520351160221518043104131447711 // 2788815009188499086581352357412492142272 The first harmonic number to exceed the following integers is: integer = 1 -> n = 2 -> harmonic number = 1.5 (to 6dp) integer = 2 -> n = 4 -> harmonic number = 2.083333 (to 6dp) integer = 3 -> n = 11 -> harmonic number = 3.019877 (to 6dp) integer = 4 -> n = 31 -> harmonic number = 4.027245 (to 6dp) integer = 5 -> n = 83 -> harmonic number = 5.002068 (to 6dp) integer = 6 -> n = 227 -> harmonic number = 6.004366 (to 6dp) integer = 7 -> n = 616 -> harmonic number = 7.001274 (to 6dp) integer = 8 -> n = 1674 -> harmonic number = 8.000485 (to 6dp) integer = 9 -> n = 4550 -> harmonic number = 9.000208 (to 6dp) integer = 10 -> n = 12367 -> harmonic number = 10.000043 (to 6dp)
Julia
const memoizer = [BigFloat(1.0), BigFloat(1.5)]
"""
harmonic(n::Integer)::BigFloat
Calculates harmonic numbers. The integer argument `n` should be positive.
"""
function harmonic(n::Integer)::BigFloat
if n < 0
throw(DomainError(n))
elseif n == 0
return BigFloat(0.0) # by convention
elseif length(memoizer) >= n
return memoizer[n]
elseif length(memoizer) + 1 == n
h = memoizer[end] + BigFloat(1.0) / n
push!(memoizer, h)
return h
elseif n < 1_000_000
start, x = length(memoizer), memoizer[end]
for i in start+1:n
push!(memoizer, (x += big"1.0" / i))
end
return memoizer[end]
else
# use H(n) = eulergamma + digamma(n + 1), instead, if memory use of memoization too large
x = n + big"1.0"
digam = BigFloat()
ccall((:mpfr_digamma, :libmpfr), Int32, (Ref{BigFloat}, Ref{BigFloat}, Int32), digam, x, 1)
return Base.MathConstants.eulergamma + digam
end
end
function testharmonics(upperlimit = 11)
n = 1
while (h = harmonic(n)) < upperlimit
nextintegerfloor = h < 1.8 ? h > 1.0 : floor(h) > floor(memoizer[n - 1])
if n < 21 || nextintegerfloor
println("harmonic($n) = $h")
nextintegerfloor && println(" $n is also the term number for the first harmonic > $(floor(h))")
end
n += 1
end
end
testharmonics()
- Output:
harmonic(1) = 1.0 harmonic(2) = 1.5 2 is also the term number for the first harmonic > 1.0 harmonic(3) = 1.833333333333333333333333333333333333333333333333333333333333333333333333333339 harmonic(4) = 2.083333333333333333333333333333333333333333333333333333333333333333333333333356 4 is also the term number for the first harmonic > 2.0 harmonic(5) = 2.283333333333333333333333333333333333333333333333333333333333333333333333333363 harmonic(6) = 2.450000000000000000000000000000000000000000000000000000000000000000000000000041 harmonic(7) = 2.592857142857142857142857142857142857142857142857142857142857142857142857142913 harmonic(8) = 2.717857142857142857142857142857142857142857142857142857142857142857142857142913 harmonic(9) = 2.828968253968253968253968253968253968253968253968253968253968253968253968254009 harmonic(10) = 2.928968253968253968253968253968253968253968253968253968253968253968253968253995 harmonic(11) = 3.019877344877344877344877344877344877344877344877344877344877344877344877344889 11 is also the term number for the first harmonic > 3.0 harmonic(12) = 3.103210678210678210678210678210678210678210678210678210678210678210678210678211 harmonic(13) = 3.180133755133755133755133755133755133755133755133755133755133755133755133755123 harmonic(14) = 3.251562326562326562326562326562326562326562326562326562326562326562326562326542 harmonic(15) = 3.318228993228993228993228993228993228993228993228993228993228993228993228993199 harmonic(16) = 3.380728993228993228993228993228993228993228993228993228993228993228993228993199 harmonic(17) = 3.439552522640757934875581934405463817228523110876052052522640757934875581934384 harmonic(18) = 3.495108078196313490431137489961019372784078666431607608078196313490431137489932 harmonic(19) = 3.547739657143681911483769068908387793836710245378976029130827892437799558542556 harmonic(20) = 3.597739657143681911483769068908387793836710245378976029130827892437799558542549 harmonic(31) = 4.027245195436520102759838180253409570739320924649712368107240380481568735938418 31 is also the term number for the first harmonic > 4.0 harmonic(83) = 5.002068272680166053728324750753870264345455215566438587478989543061001039767003 83 is also the term number for the first harmonic > 5.0 harmonic(227) = 6.004366708345566023376436217157408474650893771305512336984772241757969069086895 227 is also the term number for the first harmonic > 6.0 harmonic(616) = 7.001274097134160381487068933022945074864048309674852535721112060499845844673362 616 is also the term number for the first harmonic > 7.0 harmonic(1674) = 8.000485571995779067790304796519697445800341927883408389172647367923220595045883 1674 is also the term number for the first harmonic > 8.0 harmonic(4550) = 9.000208062931140339164179501268928624268799275400095995932594006439583360896694 4550 is also the term number for the first harmonic > 9.0 harmonic(12367) = 10.00004300827580769470675707492981720768686887243344211163998834649135547210551 12367 is also the term number for the first harmonic > 10.0
Using rationals
const harmonics = accumulate((x, y) -> x + big"1" // y, 1:12370)
println("First twenty harmonic numbers as rationals:")
foreach(i -> println(rpad(i, 3), " => ", harmonics[i]), 1:20)
println("\nThe 100th harmonic is: ", harmonics[100], "\n")
for n in 1:10
idx = findfirst(x -> x > n, harmonics)
print("First Harmonic > $n is at position $idx and is: ", harmonics[idx], "\n\n")
end
- Output:
First twenty harmonic numbers as rationals: 1 => 1//1 2 => 3//2 3 => 11//6 4 => 25//12 5 => 137//60 6 => 49//20 7 => 363//140 8 => 761//280 9 => 7129//2520 10 => 7381//2520 11 => 83711//27720 12 => 86021//27720 13 => 1145993//360360 14 => 1171733//360360 15 => 1195757//360360 16 => 2436559//720720 17 => 42142223//12252240 18 => 14274301//4084080 19 => 275295799//77597520 20 => 55835135//15519504 The 100th harmonic is: 14466636279520351160221518043104131447711//2788815009188499086581352357412492142272 First Harmonic > 1 is at position 2 and is: 3//2 First Harmonic > 2 is at position 4 and is: 25//12 First Harmonic > 3 is at position 11 and is: 83711//27720 First Harmonic > 4 is at position 31 and is: 290774257297357//72201776446800 First Harmonic > 5 is at position 83 and is: 3672441655127796364812512959533039359//734184632222154704090370027645633600 First Harmonic > 6 is at position 227 and is: 7210530454341478178114292924106791866448071719960766673184657267908514585008387695857601640547547//1200881092808579751109445892858157237623011602251376919557525378451885327053551694768211209584000 First Harmonic > 7 is at position 616 and is: 32418148234584438506244669620559801586453637524407441287287956498408167324384836135154335608450103705554364675635095796463552565723833053990601655182066284053456674137000162471237344506126617371216079120095971345235097691744446098739930413505848076716148051590567//4630321250792651667959778962730272725135972962879403462498154236164728708558353376685441262094879300811505205446228182535680317471673227192574715015221837802633281044802046825192844588381756912000006314261374254853934981337617963082619967597058012665881508170240 First Harmonic > 8 is at position 1674 and is: 138141199730356031917751723608451630140926572041533266161867229808423134343694448776284861362361049658120307218628922472967074569643922410693236711646552594017799177544898923160715275467074773052672543691485042344006328346792717577331346270889819765648723902241194752204176829233330048155240810379245143159803746553605910654524991868894244655125224374793405027930983653080599361646248333997230466686371943554425849838679719632376766617639339476485319908534320267241696294676687857612211224929536683459738832437894009105247288571220395942120501422698013842452465990861198601652372525840473807472136998877762025722745653584203415416989419985647929410780044991971375219261681255528981314607252039408935696726675566989805504130221402879//17266601943998720215934555231097205020805360283925021776131771421603665580550992851730546192114435876915764081799263950036774496079790573533431946151195888729825227967898856935376506079740151630837994958679444177020560516804785312006925540853535681555388532126434276232761609235943363516882249874992501288848101663246418034482830782765799910587627015490435618612817485429045075697874945013812237906815356826216080083918931352849328226757321069450237292900402036556169613799106635542806391854137317272454464723030848621364955770185788658456980359320481573473968897297574022422610618929844739054324826111937140267616883649616842817036401014150739236179076410461198210703989272527503945999049527912447331310263532711648780174245760000 First Harmonic > 9 is at position 4550 and is: 2803922057204811543989535496612889221797101689401339733425779071893883726062730194418722759045899108563649222513327676111056373937430194191361767957531399947657882950860187200723323262653682200418079207248717851019834955117510807290247517344853824182604853734193915236890382944826941254750728989773414206859570065960640186792890664839103798433780558510849879082382309565191620282894846917562664482131202291813204472491830923673050493632880376511556753354676402403084278296415856939754434734260623967677761451430096042117604187063669249143393076890252523001247824132437003474131230135624887874799541971404156392012052221712958314528322355541629626096561834856317166367946519841625273109689443740339231545297896759513984852874078072318627511091395704617775835151474534768408675032414448963002972590468720049363107497714657975180640114587539853156735035634359134035816625004885128358508857789053399591546078553132392557669776329295754833648165907192091248381853554990310275294735796601315160268572473977239894244510958924644461475601889585298550864600912608515846981406115574447937712467306028124552530287697424685820096894634400504911945644842525080877704081133701601474836419212357427281193595545654309059483619700606698188346807113314394305087007324692154551457096864441413765832492015835459544797423019196304195496229963766867559041682555605485756372169731071238590755894190062578370776223859945908951232125318924762539326986977852885933136208484015119262988152215227210365689996539057011922584498047033647706625149283085817504520106280560129638082705378996661011957957116006944584486439354147266379272818775057457727700062964206158298239058238442033871955799108732891594547873095023995742714560400884479305635074935996707008596929664779547461570776674459690821497530879010144224813067523793975748941040890361283562073364309054362072137372458221297707667641071756676535258762295067395479041974831597965979613081474695278123265026663091006352059826017007472704096220374679625403//311539693038123496722215310551186904103082524726970170658802584453586817702845967012058559851648961565162714074357248615113237727615741409397613403942762027416418250197869235139807433555974792925305500675331577565501246819843747067123329275977858482555209853394601249731116273203314723492820718920663543363350452108859588055589207296367576316485295250028686463172112871115738305282269830733309445640796833467195761172540352742868324044527033042113461149657815168691435568328007659916683562048346389048741726140068665849102873300936534794943981012153540913055621237325128266901087332680399450826342551172911601227269655973249675723840414635853583858640961466968497444812002669481333031399627561658239662030094505016308687209615676717899935366285911202891060019766201048122485091665890488010192945266722188069264271709669076333959826410108104815827793429989042734394907438794428685359256959974813630940895555145059961227687918900885009633119015801758204737016204228694445315173976166116707848897795690334687813067638452214980721143416739719469158748498888549650173439285892765248219730073273036517725939409971809083058041780264560576764115967213013380206017646943405476909576091357482868476761698005324735669002986992261514164471647621973497050651105016702049275202360233254308086080840811811270395182950866854592166254651076234420662875446908943737749367221987863265181700223872816922018831416550743452655599236620182838774266078307310527709836462300005876748274266928980195184478251265986886488170211071786897470230189921260226806813948222307759474759342997757077589856537503856731410785456607398210129322637205537554981139121822428297380197652187006595903988741596680008103907700803965881407891948670316751093348441767470344198247057046756517630261990510817072323901911147038583224901683429072731095666462995891134319532658460042039832285713339493699639220237857309737109028479215274870712938102667472652582674547746273957596211401299308669708874527010828161490844069542400000 First Harmonic > 10 is at position 12367 and is: 45345034307070335134555666635024887773984040811638642621462405132584886144532377986073358145244201027037357938621772142209616405161627868105778015683798484008416515631723931456447477569347791478527857610769841230294963993979332743283338510233568324905959949579062456226347439170271217301636734446205747735713945863253433483740897523300848534809108461545232146005948850103219707387931993351213767427376889962909120228840233148459893174855648971003785113710072840958355449134583864883502737527932996232809536817702108606704972939587420565678604933039194329667673092895550055241161559979675050092624926868871458018343441582176596250529929191074291128267400687630170072609517611336780875630097770219247592957358603597479920082804823686786983637602905120758432634792571445674234736536086607803064981671382406890757026618258218813838974120950628944981905245622673666105003761865085749235468633696285737487359622886699738348673700526205671244526918181580594955298707753497194308839658138978679642619068841057090153286541341850948526490774435583437961002123092419531093235665933120984675385509883201479274589825686372618722516148263128067561905455373547546391052207792967353055080744047776986080724800254366264659407835575877760954125271799727336398809285045211625324747901193314444326911865397319722296088570348383225778250731285584101470972417783253384871308701259088227660489254359610314686851359252857300051920723671060874215404857586898535480326782181158608408279619195590515799061476749269801594625879703128842040583394154984407245336678544432608949614158673227077737666441262935879700412919345767447983556401804802974663896313057250911344831472616803645836814569048455850429162273597559955668906516180684124175673531692230282389747164102840354665242183576801399969588541389263384471603494896180265191824808290512910947744283695895248414399524148990184306294157590456646295768057794302761262994594722655055964096754993330351814041078876315865417737202330307734602330970139055634545867680529440237996308032770048686378187392782129596233133189016852266246661061281495413831799689680798229084372131339265032436345645554714807426037605589639494475684095551532755566945308404954024645746798191900979340826809913903596772706839217198094879854114314539133185622222803259348108969761096223170124626733975997197050299930954519458027451118459941271137375313026666466736293093998795637218208416937639900402972180842804696606039337369141035805923341690099055369556837577997854655210298104810520933189746485979124018196909319459534410139406110909222206154476323155751314166239864898017483972228615274429236864538159277543058839398436445287447883114413176678000824006252470021316902405765477155539855660847221656031691805464444896245205294585896596429456743028035184934461863654010366528689573439474363008455160522485449967670027033722278013317043478614293767140874291908415829941007066088704322634788409768471256565985280476650542596002212877354416976349052617941581234194408366031879100064003856581616095600774809166667678552494014963764538137429108976853046746079327814801533505923050861435172931240483219080490333386411655709499209676409734678148401497706097394309678808363917083429809690764137065474753611515665686465587299537880762699588786806388064113473982700024684504900602299235464951536685894424056502683548413627701664149406689523427968563342971845946890142203694025409999036915672932674775076177305732160054942888943862654196099307096880517864598688559999274834885952923776254724920526071660112996620623947730034924455037553195043344076284277868014382989931974908029499035370171767959005233662215283043876585561861314525586427883264270058319450472084164619032769031551576680710841970852322393037539325119666017996420868400106357872834174841151636016380407477204298535548979650626662451521208388882826961445471585692683081515085943594719122539526092112699608963277450836088209198655644116097290500948029821534837319698118213632136665010898032500699148625910693981318114217177275819944652811575951322137901737402106063154587879423128895000245926444726344972103433121968004238984107852115590751270399146488824187698443051260706281584394068239758982021239369875085477123234398148089271884961390297847264240658363997995669574653293259007333471580117358907760396222393711792653221029880957923342959879228299043166437440244751583666970116390226900111585349332814785514272218273380355024057390176759649953398455756889742471454303755506911727925831506750404598474949928168799378739227974329175403082516852308081709012535982487502050229124471983831128472775878806304759862969459429435276021291396028499943188538938799118003305779761837824757679798705125552062627109499973531037724930204716526740123383060533255126341151138114409083882474220727550189504138353942985034294716389262234667241206934164898746082763605571675460470098269794908360324625747888989968089590921522328937339714693262302234111258577347278529070921677616792503989869186375043102196295624700036419726681106495130783471156636071196116110061360911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Mathematica/Wolfram Language
nums = HarmonicNumber[Range[15000]];
nums[[;; 20]]
LengthWhile[nums, LessEqualThan[#]] + 1 & /@ Range[10]
- Output:
{1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, 86021/27720, 1145993/360360, 1171733/360360, 1195757/360360, 2436559/720720, 42142223/12252240, 14274301/4084080, 275295799/77597520, 55835135/15519504} {2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367}
Nim
Using floats
import strformat
iterator h(): (int, float) =
## Yield the index of the term and its value.
var n = 1
var r = 0.0
while true:
r += 1 / n
yield (n, r)
inc n
echo "First 20 terms of the harmonic series:"
for (idx, val) in h():
echo &"{idx:2}: {val}"
if idx == 20: break
echo()
var target = 1.0
for (idx, val) in h():
if val > target:
echo &"Index of the first term greater than {target.int:2}: {idx}"
if target == 10: break
else: target += 1
- Output:
1: 1.0 2: 1.5 3: 1.833333333333333 4: 2.083333333333333 5: 2.283333333333333 6: 2.45 7: 2.592857142857143 8: 2.717857142857143 9: 2.828968253968254 10: 2.928968253968254 11: 3.019877344877345 12: 3.103210678210678 13: 3.180133755133755 14: 3.251562326562327 15: 3.318228993228994 16: 3.380728993228994 17: 3.439552522640758 18: 3.495108078196314 19: 3.547739657143682 20: 3.597739657143682 Index of the first term greater than 1: 2 Index of the first term greater than 2: 4 Index of the first term greater than 3: 11 Index of the first term greater than 4: 31 Index of the first term greater than 5: 83 Index of the first term greater than 6: 227 Index of the first term greater than 7: 616 Index of the first term greater than 8: 1674 Index of the first term greater than 9: 4550 Index of the first term greater than 10: 12367
Using big integers
import strformat
import bignum
iterator h(): (int, Rat) =
var n = 1
var r = newRat()
while true:
r += newRat(1, n)
yield (n, r)
inc n
echo "First 20 terms of the harmonic series:"
for (idx, val) in h():
echo &"{idx:2}: {val}"
if idx == 20: break
echo()
var target = 1
for (idx, val) in h():
if val > target:
echo &"Index of the first term greater than {target:2}: {idx}"
if target == 10: break
else: inc target
- Output:
1: 1 2: 3/2 3: 11/6 4: 25/12 5: 137/60 6: 49/20 7: 363/140 8: 761/280 9: 7129/2520 10: 7381/2520 11: 83711/27720 12: 86021/27720 13: 1145993/360360 14: 1171733/360360 15: 1195757/360360 16: 2436559/720720 17: 42142223/12252240 18: 14274301/4084080 19: 275295799/77597520 20: 55835135/15519504 Index of the first term greater than 1: 2 Index of the first term greater than 2: 4 Index of the first term greater than 3: 11 Index of the first term greater than 4: 31 Index of the first term greater than 5: 83 Index of the first term greater than 6: 227 Index of the first term greater than 7: 616 Index of the first term greater than 8: 1674 Index of the first term greater than 9: 4550 Index of the first term greater than 10: 12367
PARI/GP
h=0
for(n=1,20,h=h+1/n;print(n," ",h))
h=0; n=1
for(i=1,10,while(h<i,h=h+1/n;n=n+1);print(n-1))
Perl
use strict;
use warnings;
use feature 'say';
use Math::AnyNum ':overload';
use List::AllUtils 'firstidx';
my(@H,$n) = 0;
do { ++$n and push @H, $H[-1] + 1/$n } until $H[-1] >= 10;
shift @H;
say 'First twenty harmonic numbers as rationals:';
my $c = 0;
printf("%20s", $_) and (not ++$c%5) and print "\n" for @H[0..19];
say "\nIndex of first value (zero based):";
for my $i (1..10) {
printf " greater than %2d: %5s\n", $i, firstidx { $_ > $i } @H;
}
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 Index of first value (zero based): greater than 1: 1 greater than 2: 3 greater than 3: 10 greater than 4: 30 greater than 5: 82 greater than 6: 226 greater than 7: 615 greater than 8: 1673 greater than 9: 4549 greater than 10: 12366
Phix
requires("0.8.4") include mpfr.e integer n = 1, gn = 1, lim = iff(platform()=JS?8:10) mpq hn = mpq_init_set_si(1) sequence gt = {} puts(1,"First twenty harmonic numbers as rationals:\n") while gn<=lim do if n<=20 then printf(1,"%18s%s",{mpq_get_str(hn),iff(mod(n,5)?" ","\n")}) end if if n=100 then printf(1,"\nOne Hundredth:\n%s\n\n",{mpq_get_str(hn)}) end if if mpq_cmp_si(hn,gn)>0 then gt &= n gn += 1 end if n += 1 mpq_add_si(hn,hn,1,n) end while printf(1,"(one based) Index of first value:\n") for i=1 to length(gt) do printf(1," greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])}) end for
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 One Hundredth: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 (one based) Index of first value: greater than 1: 2 (second term) greater than 2: 4 (fourth term) greater than 3: 11 (eleventh term) greater than 4: 31 (thirty-first term) greater than 5: 83 (eighty-third term) greater than 6: 227 (two hundred and twenty-seventh term) greater than 7: 616 (six hundred and sixteenth term) greater than 8: 1,674 (one thousand, six hundred and seventy-fourth term) greater than 9: 4,550 (four thousand, five hundred and fiftieth term) greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)
(The last two entries are too slow/beyond the reach of the first version of mpfr.js, but it's quick on the desktop)
using standard floats
integer n = 1, gn = 1 atom hn = 1 sequence gt = {} puts(1,"First twenty harmonic numbers as fractions:\n") while gn<=10 do if n<=20 then printf(1,"%18.15f%s",{hn,iff(mod(n,5)?" ","\n")}) end if if n=100 then printf(1,"\nOne Hundredth: %18.15f\n\n",{hn}) end if if hn>gn then gt &= n gn += 1 end if n += 1 hn += 1/n end while printf(1,"(one based) Index of first value:\n") for i=1 to length(gt) do printf(1," greater than %2d: %,6d (%s term)\n",{i,gt[i],ordinal(gt[i])}) end for {} = wait_key()
- Output:
First twenty harmonic numbers as fractions: 1.000000000000000 1.500000000000000 1.833333333333333 2.083333333333333 2.283333333333333 2.450000000000000 2.592857142857143 2.717857142857143 2.828968253968254 2.928968253968254 3.019877344877345 3.103210678210678 3.180133755133755 3.251562326562327 3.318228993228994 3.380728993228994 3.439552522640758 3.495108078196314 3.547739657143682 3.597739657143682 One Hundredth: 5.187377517639621 (one based) Index of first value: greater than 1: 2 (second term) greater than 2: 4 (fourth term) greater than 3: 11 (eleventh term) greater than 4: 31 (thirty-first term) greater than 5: 83 (eighty-third term) greater than 6: 227 (two hundred and twenty-seventh term) greater than 7: 616 (six hundred and sixteenth term) greater than 8: 1,674 (one thousand, six hundred and seventy-fourth term) greater than 9: 4,550 (four thousand, five hundred and fiftieth term) greater than 10: 12,367 (twelve thousand, three hundred and sixty-seventh term)
Prolog
main:-
print_harmonic_series(20),
nl,
nth_harmonic_number(100, T),
Num is numerator(T),
Denom is denominator(T),
writef('100th harmonic number: %t/%t\n', [Num, Denom]),
nl,
print_first_harmonic_greater_than(10).
print_harmonic_series(N):-
writef('First %t harmonic numbers:\n', [N]),
harmonic_first(H),
print_harmonic_series(N, H).
print_harmonic_series(N, H):-
H = h(I, T),
Num is numerator(T),
Denom is denominator(T),
writef('%3r. %t/%t\n', [I, Num, Denom]),
(I == N, ! ; harmonic_next(H, H1), print_harmonic_series(N, H1)).
print_first_harmonic_greater_than(N):-
harmonic_first(H),
print_first_harmonic_greater_than(1, N, H).
print_first_harmonic_greater_than(N, L, _):-
N > L,
!.
print_first_harmonic_greater_than(N, L, H):-
H = h(P, T),
(T > N ->
writef('Position of first term >%3r: %t\n', [N, P]),
N1 is N + 1
;
N1 = N),
harmonic_next(H, H1),
print_first_harmonic_greater_than(N1, L, H1).
harmonic_first(h(1, 1)).
harmonic_next(h(N1, T1), h(N2, T2)):-
N2 is N1 + 1,
T2 is T1 + 1 rdiv N2.
nth_harmonic_number(N, T):-
harmonic_first(H),
nth_harmonic_number(N, T, H).
nth_harmonic_number(N, T, h(N, T)):-!.
nth_harmonic_number(N, T, H1):-
harmonic_next(H1, H2),
nth_harmonic_number(N, T, H2).
- Output:
First 20 harmonic numbers: 1. 1/1 2. 3/2 3. 11/6 4. 25/12 5. 137/60 6. 49/20 7. 363/140 8. 761/280 9. 7129/2520 10. 7381/2520 11. 83711/27720 12. 86021/27720 13. 1145993/360360 14. 1171733/360360 15. 1195757/360360 16. 2436559/720720 17. 42142223/12252240 18. 14274301/4084080 19. 275295799/77597520 20. 55835135/15519504 100th harmonic number: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 Position of first term > 1: 2 Position of first term > 2: 4 Position of first term > 3: 11 Position of first term > 4: 31 Position of first term > 5: 83 Position of first term > 6: 227 Position of first term > 7: 616 Position of first term > 8: 1674 Position of first term > 9: 4550 Position of first term > 10: 12367
Python
A generator function using fractions:
from fractions import Fraction
def harmonic_series():
n, h = Fraction(1), Fraction(1)
while True:
yield h
h += 1 / (n + 1)
n += 1
if __name__ == '__main__':
from itertools import islice
for n, d in (h.as_integer_ratio() for h in islice(harmonic_series(), 20)):
print(n, '/', d)
- Output:
1 / 1 3 / 2 11 / 6 25 / 12 137 / 60 49 / 20 363 / 140 761 / 280 7129 / 2520 7381 / 2520 83711 / 27720 86021 / 27720 1145993 / 360360 1171733 / 360360 1195757 / 360360 2436559 / 720720 42142223 / 12252240 14274301 / 4084080 275295799 / 77597520 55835135 / 15519504
Or alternatively, in terms of itertools.accumulate:
'''Harmonic series'''
from fractions import Fraction
from itertools import accumulate, count, islice
from operator import add
# harmonic :: [Fraction]
def harmonic():
'''Non finite stream of the terms
of the Harmonic series.
'''
return accumulate(
(1 / Fraction(x) for x in count(1)),
add
)
# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Tests of the harmonic series function'''
print('First 20 terms of the harmonic series:')
print('\n'.join([
showFraction(nd) for nd in islice(harmonic(), 20)
]))
print('\n100th term:')
print(
showFraction(
next(islice(harmonic(), 99, None))
)
)
print('')
print(
'One-based indices of terms above threshold values:'
)
indexedHarmonic = enumerate(harmonic())
print('\n'.join([
next(
showFirstLimit(n)(x) for x
in indexedHarmonic if n < x[1]
) for n in range(1, 1 + 10)
]))
# ------------------ DISPLAY FORMATTING ------------------
# showFraction :: Fraction -> String
def showFraction(nd):
'''String representation of the fraction nd.
'''
n, d = nd.as_integer_ratio()
return f'{n} / {d}'
# showFirstLimit :: Int -> (Int, Fraction) -> String
def showFirstLimit(n):
'''Report of 1-based index of first term
with a value over n
'''
def go(indexedFraction):
i = indexedFraction[0]
return f'Term {1 + i} is the first above {n}'
return go
# MAIN ---
if __name__ == '__main__':
main()
- Output:
First 20 terms of the harmonic series: 1 / 1 3 / 2 11 / 6 25 / 12 137 / 60 49 / 20 363 / 140 761 / 280 7129 / 2520 7381 / 2520 83711 / 27720 86021 / 27720 1145993 / 360360 1171733 / 360360 1195757 / 360360 2436559 / 720720 42142223 / 12252240 14274301 / 4084080 275295799 / 77597520 55835135 / 15519504 100th term: 14466636279520351160221518043104131447711 / 2788815009188499086581352357412492142272 One-based indices of terms above threshold values: Term 2 is the first above 1 Term 4 is the first above 2 Term 11 is the first above 3 Term 31 is the first above 4 Term 83 is the first above 5 Term 227 is the first above 6 Term 616 is the first above 7 Term 1674 is the first above 8 Term 4550 is the first above 9 Term 12367 is the first above 10
Quackery
[ $ "bigrat.qky" loadfile ] now!
0 n->v
20 times
[ i^ 1+ n->v 1/v v+
2dup 20 point$ echo$
say " = "
2dup vulgar$ echo$ cr ]
2drop
cr
1 temp put
0 n->v 1
[ dup dip
[ n->v 1/v v+
temp share n->v 2over v< ]
swap if
[ temp share echo
say " : "
dup echo cr
1 temp tally ]
temp share 11 < while
1+
again ]
temp release
drop 2drop
- Output:
1 = 1/1 1.5 = 3/2 1.83333333333333333333 = 11/6 2.08333333333333333333 = 25/12 2.28333333333333333333 = 137/60 2.45 = 49/20 2.59285714285714285714 = 363/140 2.71785714285714285714 = 761/280 2.82896825396825396825 = 7129/2520 2.92896825396825396825 = 7381/2520 3.01987734487734487734 = 83711/27720 3.10321067821067821068 = 86021/27720 3.18013375513375513376 = 1145993/360360 3.25156232656232656233 = 1171733/360360 3.31822899322899322899 = 1195757/360360 3.38072899322899322899 = 2436559/720720 3.43955252264075793488 = 42142223/12252240 3.49510807819631349043 = 14274301/4084080 3.54773965714368191148 = 275295799/77597520 3.59773965714368191148 = 55835135/15519504 1 : 2 2 : 4 3 : 11 4 : 31 5 : 83 6 : 227 7 : 616 8 : 1674 9 : 4550 10 : 12367
R
Direct Summation
The talk page helpfully points out that we can be remarkably lazy here.
HofN <- function(n) sum(1/seq_len(n)) #Task 1
H <- sapply(1:100000, HofN)
print(H[1:20]) #Task 2
print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch
- Output:
> print(H[1:20]) #Task 2 [1] 1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968 3.019877 3.103211 3.180134 3.251562 [15] 3.318229 3.380729 3.439553 3.495108 3.547740 3.597740 > print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch [1] 2 4 11 31 83 227 616 1674 4550 12367
Cumulative Sums
As for doing this properly, R provides a handy cumsum function.
firstNHarmonicNumbers <- function(n) cumsum(1/seq_len(n)) #Task 1
H <- firstNHarmonicNumbers(100000) #Runs stunningly quick
print(H[1:20]) #Task 2
print(sapply(1:10, function(x) which.max(H > x))) #Task 3 and stretch
Raku
Using Lingua::EN::Numbers from the Raku ecosystem.
use Lingua::EN::Numbers;
my @H = [\+] (1..*).map: { FatRat.new: 1, $_ };
say "First twenty harmonic numbers as rationals:\n",
@H[^20]».&pretty-rat.batch(5)».fmt("%18s").join: "\n";
put "\nOne Hundredth:\n", pretty-rat @H[99];
say "\n(zero based) Index of first value:";
printf " greater than %2d: %6s (%s term)\n",
$_, comma( my $i = @H.first(* > $_, :k) ), ordinal 1 + $i for 1..10;
- Output:
First twenty harmonic numbers as rationals: 1 3/2 11/6 25/12 137/60 49/20 363/140 761/280 7129/2520 7381/2520 83711/27720 86021/27720 1145993/360360 1171733/360360 1195757/360360 2436559/720720 42142223/12252240 14274301/4084080 275295799/77597520 55835135/15519504 One Hundredth: 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 (zero based) Index of first value: greater than 1: 1 (second term) greater than 2: 3 (fourth term) greater than 3: 10 (eleventh term) greater than 4: 30 (thirty-first term) greater than 5: 82 (eighty-third term) greater than 6: 226 (two hundred twenty-seventh term) greater than 7: 615 (six hundred sixteenth term) greater than 8: 1,673 (one thousand, six hundred seventy-fourth term) greater than 9: 4,549 (four thousand, five hundred fiftieth term) greater than 10: 12,366 (twelve thousand, three hundred sixty-seventh term)
REXX
The default number of decimal digits (9) could've been used instead of 80 for this task's particular limits.
/*REXX pgm to calculate N numbers (sums) in the harmonic series and also when they > X. */
parse arg digs sums high ints /*obtain optional arguments from the CL*/
if digs='' | digs="," then digs= 80 /*Not specified? Then use the default.*/
if sums='' | sums="," then sums= 20 /* " " " " " " */
if high='' | high="," then high= 10 /* " " " " " " */
if ints='' | ints="," then ints= 1 2 3 4 5 6 7 8 9 10 /*Not specified? " " " */
w= length(sums) + 2 /*width of Nth harmonic index + suffix.*/
numeric digits digs /*have REXX use more numeric dec. digs.*/
s= 0 /*initialize harmonic series sum to 0. */
do j=1 for sums; s= s + 1/j /*calc "sums" of harmonic series nums.*/
@iter= right((j)th(j), w) /*obtain a nicely formatted sum index. */
say right(@iter, w) 'harmonic sum ──►' s /*indent the output to the terminal. */
end /*j*/
say /*have a blank line between output sets*/
many= words(ints) /*obtain number of limits to be used. */
z= word(ints, 1) /* " the first " " " " */
lastInt= word(ints, many) /* " " last " " " " */
w= length(lastInt) /*W: is the maximum width of any limit*/
#= 1 /*a pointer to a list of integer limits*/
s= 0 /*initialize harmonic series sum to 0. */
do j=1; s= s + 1/j /*calculate sums of harmonic sum index.*/
if s<=z then iterate /*Is sum <= a limit? Then keep going. */
iter= commas(j)th(j) /*obtain a nicely formatted sum index. */
L= length(iter) /*obtain length so as to align output. */
@iter= right(iter, max(L, 25) ) /*indent the output to the terminal. */
say @iter " iteration of the harmonic series, the sum is greater than " right(z, w)
#= # + 1 /*bump the pointer to the next limit. */
if #>many then leave /*Are at the end of the limits? Done. */
z= word(ints, #) /*point to the next limit to be used. */
end /*j*/ /* [↑] above indices are unity─based. */
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?
th: parse arg x; return word('th st nd rd', 1 + (x//10) *(x//100%10\==1) *(x//10<4))
- output when using the default inputs:
1st harmonic sum ──► 1 2nd harmonic sum ──► 1.5 3rd harmonic sum ──► 1.8333333333333333333333333333333333333333333333333333333333333333333333333333333 4th harmonic sum ──► 2.0833333333333333333333333333333333333333333333333333333333333333333333333333333 5th harmonic sum ──► 2.2833333333333333333333333333333333333333333333333333333333333333333333333333333 6th harmonic sum ──► 2.4500000000000000000000000000000000000000000000000000000000000000000000000000000 7th harmonic sum ──► 2.5928571428571428571428571428571428571428571428571428571428571428571428571428571 8th harmonic sum ──► 2.7178571428571428571428571428571428571428571428571428571428571428571428571428571 9th harmonic sum ──► 2.8289682539682539682539682539682539682539682539682539682539682539682539682539682 10th harmonic sum ──► 2.9289682539682539682539682539682539682539682539682539682539682539682539682539682 11th harmonic sum ──► 3.0198773448773448773448773448773448773448773448773448773448773448773448773448773 12th harmonic sum ──► 3.1032106782106782106782106782106782106782106782106782106782106782106782106782106 13th harmonic sum ──► 3.1801337551337551337551337551337551337551337551337551337551337551337551337551337 14th harmonic sum ──► 3.2515623265623265623265623265623265623265623265623265623265623265623265623265623 15th harmonic sum ──► 3.3182289932289932289932289932289932289932289932289932289932289932289932289932290 16th harmonic sum ──► 3.3807289932289932289932289932289932289932289932289932289932289932289932289932290 17th harmonic sum ──► 3.4395525226407579348755819344054638172285231108760520525226407579348755819344055 18th harmonic sum ──► 3.4951080781963134904311374899610193727840786664316076080781963134904311374899611 19th harmonic sum ──► 3.5477396571436819114837690689083877938367102453789760291308278924377995585425927 20th harmonic sum ──► 3.5977396571436819114837690689083877938367102453789760291308278924377995585425927 2nd iteration of the harmonic series, the sum is greater than 1 4th iteration of the harmonic series, the sum is greater than 2 11th iteration of the harmonic series, the sum is greater than 3 31st iteration of the harmonic series, the sum is greater than 4 83rd iteration of the harmonic series, the sum is greater than 5 227th iteration of the harmonic series, the sum is greater than 6 616th iteration of the harmonic series, the sum is greater than 7 1,674th iteration of the harmonic series, the sum is greater than 8 4,550th iteration of the harmonic series, the sum is greater than 9 12,367th iteration of the harmonic series, the sum is greater than 10
Ring
decimals(12)
sum = 0
nNew = 1
limit = 13000
Harmonic = []
for n = 1 to limit
sum += 1/n
add(Harmonic,[n,sum])
next
see "The first twenty harmonic numbers are:" + nl
for n = 1 to 20
see "" + Harmonic[n][1] + " -> " + Harmonic[n][2] + nl
next
see nl
for m = 1 to 10
for n = nNew to len(Harmonic)
if Harmonic[n][2] > m
see "The first harmonic number greater than "
see "" + m + " is " + Harmonic[n][2] + ", at position " + n + nl
nNew = n
exit
ok
next
next
- Output:
The first twenty harmonic numbers are: 1 -> 1 2 -> 1.500000000000 3 -> 1.833333333333 4 -> 2.083333333333 5 -> 2.283333333333 6 -> 2.450000000000 7 -> 2.592857142857 8 -> 2.717857142857 9 -> 2.828968253968 10 -> 2.928968253968 11 -> 3.019877344877 12 -> 3.103210678211 13 -> 3.180133755134 14 -> 3.251562326562 15 -> 3.318228993229 16 -> 3.380728993229 17 -> 3.439552522641 18 -> 3.495108078196 19 -> 3.547739657144 20 -> 3.597739657144 The first harmonic number greater than 1 is 1.500000000000, at position 2 The first harmonic number greater than 2 is 2.083333333333, at position 4 The first harmonic number greater than 3 is 3.019877344877, at position 11 The first harmonic number greater than 4 is 4.027245195437, at position 31 The first harmonic number greater than 5 is 5.002068272680, at position 83 The first harmonic number greater than 6 is 6.004366708346, at position 227 The first harmonic number greater than 7 is 7.001274097134, at position 616 The first harmonic number greater than 8 is 8.000485571996, at position 1674 The first harmonic number greater than 9 is 9.000208062931, at position 4550 The first harmonic number greater than 10 is 10.000043008276, at position 12367
Rust
Using big rationals and big integers from the num crate.
use num::rational::Ratio;
use num::{BigInt, FromPrimitive};
fn main() {
for n in 1..=20 {
println!("Harmonic number {} = {}", n, harmonic_number(n.into()));
}
println!("Harmonic number 100 = {}", harmonic_number(100.into()));
let max = 5;
let mut target = 1;
let mut i = 1;
while target <= max {
if harmonic_number(i.into()) > Ratio::from_integer(FromPrimitive::from_u64(target).unwrap())
{
println!("Position of first term > {} is {}", target, i);
target += 1;
}
i += 1;
}
}
fn harmonic_number(n: BigInt) -> Ratio<BigInt> {
let mut result: Ratio<BigInt> = Ratio::from_integer(FromPrimitive::from_u8(0).unwrap());
let mut i: BigInt = FromPrimitive::from_u8(1).unwrap();
let one: Ratio<BigInt> = Ratio::from_integer(FromPrimitive::from_u8(1).unwrap());
while i <= n {
result = &result + &one / &i;
i += 1;
}
result
}
- Output:
Harmonic number 1 = 1 Harmonic number 2 = 3/2 Harmonic number 3 = 11/6 Harmonic number 4 = 25/12 Harmonic number 5 = 137/60 Harmonic number 6 = 49/20 Harmonic number 7 = 363/140 Harmonic number 8 = 761/280 Harmonic number 9 = 7129/2520 Harmonic number 10 = 7381/2520 Harmonic number 11 = 83711/27720 Harmonic number 12 = 86021/27720 Harmonic number 13 = 1145993/360360 Harmonic number 14 = 1171733/360360 Harmonic number 15 = 1195757/360360 Harmonic number 16 = 2436559/720720 Harmonic number 17 = 42142223/12252240 Harmonic number 18 = 14274301/4084080 Harmonic number 19 = 275295799/77597520 Harmonic number 20 = 55835135/15519504 Harmonic number 100 = 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 Position of first term > 1 is 2 Position of first term > 2 is 4 Position of first term > 3 is 11 Position of first term > 4 is 31 Position of first term > 5 is 83
Verilog
module main;
integer n, i;
real h;
initial begin
h = 0.0;
$display("The first twenty harmonic numbers are:");
for(n=1; n<=20; n=n+1) begin
h = h + 1.0 / n;
$display(n, " ", h);
end
$display("");
h = 1.0;
n = 2;
for(i=2; i<=10; i=i+1) begin
while (h < i) begin
h = h + 1.0 / n;
n = n + 1;
end
$write("The first harmonic number greater than ");
$display(i, " is ", h, ", at position ", n-1);
end
$finish ;
end
endmodule
Wren
import "/big" for BigRat
import "/fmt" for Fmt
var harmonic = Fn.new { |n| (1..n).reduce(BigRat.zero) { |sum, i| sum + BigRat.one/i } }
BigRat.showAsInt = true
System.print("The first 20 harmonic numbers and the 100th, expressed in rational form, are:")
var numbers = (1..20).toList
numbers.add(100)
for (i in numbers) Fmt.print("$3d : $s", i, harmonic.call(i))
System.print("\nThe first harmonic number to exceed the following integers is:")
var i = 1
var limit = 10
var n = 1
var h = 0
while (true) {
h = h + 1/n
if (h > i) {
Fmt.print("integer = $2d -> n = $,6d -> harmonic number = $9.6f (to 6dp)", i, n, h)
i = i + 1
if (i > limit) return
}
n = n + 1
}
- Output:
The first 20 harmonic numbers and the 100th, expressed in rational form, are: 1 : 1 2 : 3/2 3 : 11/6 4 : 25/12 5 : 137/60 6 : 49/20 7 : 363/140 8 : 761/280 9 : 7129/2520 10 : 7381/2520 11 : 83711/27720 12 : 86021/27720 13 : 1145993/360360 14 : 1171733/360360 15 : 1195757/360360 16 : 2436559/720720 17 : 42142223/12252240 18 : 14274301/4084080 19 : 275295799/77597520 20 : 55835135/15519504 100 : 14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 The first harmonic number to exceed the following integers is: integer = 1 -> n = 2 -> harmonic number = 1.500000 (to 6dp) integer = 2 -> n = 4 -> harmonic number = 2.083333 (to 6dp) integer = 3 -> n = 11 -> harmonic number = 3.019877 (to 6dp) integer = 4 -> n = 31 -> harmonic number = 4.027245 (to 6dp) integer = 5 -> n = 83 -> harmonic number = 5.002068 (to 6dp) integer = 6 -> n = 227 -> harmonic number = 6.004367 (to 6dp) integer = 7 -> n = 616 -> harmonic number = 7.001274 (to 6dp) integer = 8 -> n = 1,674 -> harmonic number = 8.000486 (to 6dp) integer = 9 -> n = 4,550 -> harmonic number = 9.000208 (to 6dp) integer = 10 -> n = 12,367 -> harmonic number = 10.000043 (to 6dp)
XPL0
func real Harmonic(N); \Return Nth harmonic number
int N; real X;
[X:= 1.0;
while N >= 2 do
[X:= X + 1.0/float(N); N:= N-1];
return X;
];
int N, M;
[for N:= 1 to 20 do
[RlOut(0, Harmonic(N));
if rem(N/5) = 0 then CrLf(0);
];
for M:= 1 to 10 do
[N:= 1;
repeat N:= N+1 until Harmonic(N) > float(M);
IntOut(0, M);
Text(0, ": ");
IntOut(0, N);
CrLf(0);
];
]
- Output:
1.00000 1.50000 1.83333 2.08333 2.28333 2.45000 2.59286 2.71786 2.82897 2.92897 3.01988 3.10321 3.18013 3.25156 3.31823 3.38073 3.43955 3.49511 3.54774 3.59774 1: 2 2: 4 3: 11 4: 31 5: 83 6: 227 7: 616 8: 1674 9: 4550 10: 12367