Hickerson series of almost integers
You are encouraged to solve this task according to the task description, using any language you may know.
The following function, due to D. Hickerson, is said to generate "Almost integers" by the
"Almost Integer" page of Wolfram MathWorld, (December 31 2013). (See formula numbered 51.)
The function is:
It is said to produce "almost integers" for n between 1 and 17.
The purpose of the task is to verify this assertion.
Assume that an "almost integer" has either a nine or a zero as its first digit after the decimal point of its decimal string representation
- Task
Calculate all values of the function checking and stating which are "almost integers".
Note: Use extended/arbitrary precision numbers in your calculation if necessary to ensure you have adequate precision of results as for example:
h(18) = 3385534663256845326.39...
Ada
<lang Ada>with Ada.Text_IO; with Ada.Numerics.Generic_Elementary_Functions;
procedure Almost_Integers is
type Real is new Long_Long_Float;
package Real_IO is
new Ada.Text_IO.Float_IO (Real);
package Integer_IO is
new Ada.Text_IO.Integer_IO (Integer);
function Faculty (N : in Long_Long_Integer) return Long_Long_Integer is
(if N < 2 then N else N * Faculty (N - 1));
function Hickerson (N : in Integer) return Real
is
package Math is
new Ada.Numerics.Generic_Elementary_Functions (Real);
LN2 : constant Real := Math.Log (2.0, Base => Ada.Numerics.E);
Numerator : constant Real := Real (Faculty (Long_Long_Integer (N)));
Denominator : constant Real := 2.0 * LN2 ** (N + 1);
begin
return Numerator / Denominator;
end Hickerson;
function Is_Almost_Integer (N : Real) return Boolean is
Image : String (1 .. 100);
begin
Real_IO.Put (Image, N, Exp => 0, Aft => 2);
pragma Assert (Image (Image'Last - 2) = '.');
case Image (Image'Last - 1) is
when '0' | '9' => return True;
when others => return False;
end case;
end Is_Almost_Integer;
use Ada.Text_IO; Placeholder : String := " n h(n) almost"; Image_N : String renames Placeholder ( 1 .. 2); Image_H : String renames Placeholder ( 4 .. 31); Image_A : String renames Placeholder (34 .. 39);
begin
Put_Line (Placeholder); Image_N := (others => '-'); Image_H := (others => '-'); Image_A := (others => '-'); Put_Line (Placeholder);
for N in 1 .. 17 loop
declare
H : constant Real := Hickerson (N);
I : constant Boolean := Is_Almost_Integer (H);
begin
Integer_IO.Put (Image_N, N);
Real_IO.Put (Image_H, H, Exp => 0, Aft => 4);
Image_A := (if I then "TRUE " else "FALSE");
Put_Line (Placeholder);
end;
end loop;
end Almost_Integers;</lang>
- Output:
n h(n) almost -- ---------------------------- ------ 1 1.0407 TRUE 2 3.0028 TRUE 3 12.9963 TRUE 4 74.9987 TRUE 5 541.0015 TRUE 6 4683.0012 TRUE 7 47292.9987 TRUE 8 545834.9979 TRUE 9 7087261.0016 TRUE 10 102247563.0053 TRUE 11 1622632572.9976 TRUE 12 28091567594.9816 TRUE 13 526858348381.0012 TRUE 14 10641342970443.0845 TRUE 15 230283190977853.0370 TRUE 16 5315654681981354.5100 FALSE 17 130370767029135900.0000 TRUE
ALGOL 68
<lang algol68># determine whether the first few Hickerson numbers really are "near integers" #
- The Hickerson number n is defined by: h(n) = n! / ( 2 * ( (ln 2)^(n+1) ) ) #
- so: h(1) = 1 / ( 2 * ( ( ln 2 ) ^ 2 ) #
- and: h(n) = ( n / ln 2 ) * h(n-1) #
- set the precision of LONG LONG numbers #
PR precision 100 PR
- calculate the Hickerson numbers #
LONG LONG REAL ln2 = long long ln( 2 ); [ 1 : 18 ]LONG LONG REAL h;
h[ 1 ] := 0.5 / ( ln2 * ln2 ); FOR n FROM 2 TO UPB h DO
h[ n ] := ( n * h[ n - 1 ] ) / ln2
OD;
- determine the first digit after the point in each h(n) - if it is 0 or 9 #
- the number is a "near integer" #
FOR n TO UPB h DO
INT first decimal = SHORTEN SHORTEN ( ( ENTIER ( h[ n ] * 10 ) ) MOD 10 );
print( ( whole( n, -4 )
, " "
, fixed( h[ n ], 40, 4 )
, IF first decimal = 0 OR first decimal = 9
THEN
" a near integer"
ELSE
" NOT a near integer"
FI
, newline
)
)
OD</lang>
- Output:
1 +1.0407 a near integer 2 +3.0028 a near integer 3 +12.9963 a near integer 4 +74.9987 a near integer 5 +541.0015 a near integer 6 +4683.0012 a near integer 7 +47292.9987 a near integer 8 +545834.9979 a near integer 9 +7087261.0016 a near integer 10 +102247563.0053 a near integer 11 +1622632572.9976 a near integer 12 +28091567594.9816 a near integer 13 +526858348381.0012 a near integer 14 +10641342970443.0845 a near integer 15 +230283190977853.0374 a near integer 16 +5315654681981354.5131 NOT a near integer 17 +130370767029135900.4580 NOT a near integer 18 +3385534663256845326.3903 NOT a near integer
AWK
<lang AWK>
- syntax: GAWK -M -f HICKERSON_SERIES_OF_ALMOST_INTEGERS.AWK
- using GNU Awk 4.1.0, API: 1.0 (GNU MPFR 3.1.2, GNU MP 5.1.2)
BEGIN {
PREC = 100
for (i=1; i<=17; i++) {
h = sprintf("%25.5f",factorial(i) / (2 * log(2) ^ (i + 1)))
msg = (h ~ /\.[09]/) ? "true" : "false"
printf("%2d %s almost integer: %s\n",i,h,msg)
}
exit(0)
} function factorial(n, i,out) {
out = 1
for (i=2; i<=n; i++) {
out *= i
}
return(out)
} </lang>
Output:
1 1.04068 almost integer: true 2 3.00278 almost integer: true 3 12.99629 almost integer: true 4 74.99874 almost integer: true 5 541.00152 almost integer: true 6 4683.00125 almost integer: true 7 47292.99873 almost integer: true 8 545834.99791 almost integer: true 9 7087261.00162 almost integer: true 10 102247563.00527 almost integer: true 11 1622632572.99755 almost integer: true 12 28091567594.98157 almost integer: true 13 526858348381.00125 almost integer: true 14 10641342970443.08453 almost integer: true 15 230283190977853.03744 almost integer: true 16 5315654681981354.51308 almost integer: false 17 130370767029135900.45799 almost integer: false
Bracmat
This solution approximates with enough terms to make the first two decimals right with high likelyhood. (It stops adding more terms when the first three decimals in two consecutive approximations are the same.) See Natural logarithm of 2
Bracmat has no built in support for fixed point notation of rational numbers. Therefore each Hickerson number is split in an integer part (using integer division) and a decimal part (using a string operation) before outputting.
<lang bracmat>( 0:?n & 1:?fac & whl
' ( 1+!n:~>17:?n
& !n*!fac:?fac
& -1:?k
& 0:?L2
& 0:?oldN
& whl
' ( 1+!k:?k
& ((2*!k+1)*9^!k)^-1+!L2:?L2
& !fac*1/2*(2/3*!L2)^(-1*(!n+1)):?N
& div$(1000*!N+1/2.1):?newN
& !newN:~!oldN:?oldN
)
& out
$ ( str$("h(" !n ") =")
( div$(!newN.1000):?intpart
& @(!newN:!intpart ?decimalpart)
& str$(!intpart "." !decimalpart)
)
( @(!decimalpart:(0|9) ?)
& "is an almost-integer."
| "is NOT an almost-integer."
)
)
)
);</lang>
- Output:
h(1) = 1.041 is an almost-integer. h(2) = 3.003 is an almost-integer. h(3) = 12.996 is an almost-integer. h(4) = 74.999 is an almost-integer. h(5) = 541.002 is an almost-integer. h(6) = 4683.001 is an almost-integer. h(7) = 47292.999 is an almost-integer. h(8) = 545834.998 is an almost-integer. h(9) = 7087261.002 is an almost-integer. h(10) = 102247563.005 is an almost-integer. h(11) = 1622632572.998 is an almost-integer. h(12) = 28091567594.982 is an almost-integer. h(13) = 526858348381.001 is an almost-integer. h(14) = 10641342970443.085 is an almost-integer. h(15) = 230283190977853.038 is an almost-integer. h(16) = 5315654681981354.513 is NOT an almost-integer. h(17) = 130370767029135900.458 is NOT an almost-integer.
C
<lang c>#include <stdio.h>
- include <mpfr.h>
void h(int n) { MPFR_DECL_INIT(a, 200); MPFR_DECL_INIT(b, 200);
mpfr_fac_ui(a, n, MPFR_RNDD); // a = n!
mpfr_set_ui(b, 2, MPFR_RNDD); // b = 2 mpfr_log(b, b, MPFR_RNDD); // b = log(b) mpfr_pow_ui(b, b, n + 1, MPFR_RNDD); // b = b^(n+1)
mpfr_div(a, a, b, MPFR_RNDD); // a = a / b mpfr_div_ui(a, a, 2, MPFR_RNDD); // a = a / 2
mpfr_frac(b, a, MPFR_RNDD); // b = a - [a] mpfr_printf("%2d: %23.4Rf %c\n", n, a, mpfr_cmp_d(b, .1) * mpfr_cmp_d(b, .9) > 0 ? 'Y' : 'N'); }
int main(void) { int n; for (n = 1; n <= 17; n++) h(n);
return 0; }</lang>
- Output:
1: 1.0407 Y 2: 3.0028 Y 3: 12.9963 Y 4: 74.9987 Y 5: 541.0015 Y 6: 4683.0012 Y 7: 47292.9987 Y 8: 545834.9979 Y 9: 7087261.0016 Y 10: 102247563.0053 Y 11: 1622632572.9976 Y 12: 28091567594.9816 Y 13: 526858348381.0012 Y 14: 10641342970443.0845 Y 15: 230283190977853.0374 Y 16: 5315654681981354.5131 N 17: 130370767029135900.4580 N
C++
<lang cpp>#include <iostream>
- include <iomanip>
- include <boost/multiprecision/cpp_dec_float.hpp>
- include <boost/math/constants/constants.hpp>
typedef boost::multiprecision::cpp_dec_float_50 decfloat;
int main() {
const decfloat ln_two = boost::math::constants::ln_two<decfloat>();
decfloat numerator = 1, denominator = ln_two;
for(int n = 1; n <= 17; n++) {
decfloat h = (numerator *= n) / (denominator *= ln_two) / 2;
decfloat tenths_dig = floor((h - floor(h)) * 10);
std::cout << "h(" << std::setw(2) << n << ") = " << std::setw(25) << std::fixed << h <<
(tenths_dig == 0 || tenths_dig == 9 ? " is " : " is NOT ") << "an almost-integer.\n";
}
} </lang>
- Output:
h( 1) = 1.040684 is an almost-integer. h( 2) = 3.002781 is an almost-integer. h( 3) = 12.996291 is an almost-integer. h( 4) = 74.998735 is an almost-integer. h( 5) = 541.001519 is an almost-integer. h( 6) = 4683.001247 is an almost-integer. h( 7) = 47292.998731 is an almost-integer. h( 8) = 545834.997907 is an almost-integer. h( 9) = 7087261.001623 is an almost-integer. h(10) = 102247563.005271 is an almost-integer. h(11) = 1622632572.997550 is an almost-integer. h(12) = 28091567594.981572 is an almost-integer. h(13) = 526858348381.001248 is an almost-integer. h(14) = 10641342970443.084532 is an almost-integer. h(15) = 230283190977853.037436 is an almost-integer. h(16) = 5315654681981354.513077 is NOT an almost-integer. h(17) = 130370767029135900.457985 is NOT an almost-integer.
Clojure
In order to get enough precision, the natural logarithm of 2 had to be entered manually; the BigDecimal implementation does not have a high-precision logarithm function.
<lang clojure>(defn hickerson
"Hickerson number, calculated with BigDecimals and manually-entered high-precision value for ln(2)."
[n]
(let [n! (apply *' (range 1M (inc n)))]
(.divide n! (*' 2 (.pow 0.693147180559945309417232121458M (inc n)))
30 BigDecimal/ROUND_HALF_UP)))
(defn almost-integer?
"Tests whether the first digit after the decimal is 0 or 9."
[x]
(let [first-digit (int (mod (.divide (*' x 10) 1.0M 0 BigDecimal/ROUND_DOWN)
10))]
(or (= 0 first-digit) (= 9 first-digit))))
- Execute for side effects
(doseq [n (range 1 18) :let [h (hickerson n)]]
(println (format "%2d %24.5f" n h)
(if (almost-integer? h)
"almost integer"
"NOT almost integer")))</lang>
- Output:
1 1.04068 almost integer 2 3.00278 almost integer 3 12.99629 almost integer 4 74.99874 almost integer 5 541.00152 almost integer 6 4683.00125 almost integer 7 47292.99873 almost integer 8 545834.99791 almost integer 9 7087261.00162 almost integer 10 102247563.00527 almost integer 11 1622632572.99755 almost integer 12 28091567594.98157 almost integer 13 526858348381.00125 almost integer 14 10641342970443.08453 almost integer 15 230283190977853.03744 almost integer 16 5315654681981354.51308 NOT almost integer 17 130370767029135900.45799 NOT almost integer
COBOL
<lang cobol> >>SOURCE FREE IDENTIFICATION DIVISION. PROGRAM-ID. hickerson-series.
ENVIRONMENT DIVISION. CONFIGURATION SECTION. REPOSITORY.
FUNCTION ALL INTRINSIC .
DATA DIVISION. WORKING-STORAGE SECTION. 01 n PIC 99 COMP.
01 h PIC Z(19)9.9(10).
01 First-Decimal-Digit-Pos CONSTANT 22.
PROCEDURE DIVISION.
PERFORM VARYING n FROM 0 BY 1 UNTIL n > 17
COMPUTE h = FACTORIAL(n) / (2 * LOG(2) ** (n + 1))
DISPLAY "h(" n ") = " h " which is " NO ADVANCING
IF h (First-Decimal-Digit-Pos:1) = "0" OR "9"
DISPLAY "an almost integer."
ELSE
DISPLAY "not an almost integer."
END-IF
END-PERFORM
.
END PROGRAM hickerson-series.</lang>
- Output:
h(00) = 0.7213475204 which is not an almost integer. h(01) = 1.0406844905 which is an almost integer. h(02) = 3.0027807071 which is an almost integer. h(03) = 12.9962905052 which is an almost integer. h(04) = 74.9987354476 which is an almost integer. h(05) = 541.0015185164 which is an almost integer. h(06) = 4683.0012472622 which is an almost integer. h(07) = 47292.9987313146 which is an almost integer. h(08) = 545834.9979074851 which is an almost integer. h(09) = 7087261.0016228991 which is an almost integer. h(10) = 102247563.0052710420 which is an almost integer. h(11) = 1622632572.9975500498 which is an almost integer. h(12) = 28091567594.9815724407 which is an almost integer. h(13) = 526858348381.0012482861 which is an almost integer. h(14) = 10641342970443.0845319270 which is an almost integer. h(15) = 230283190977853.0374360391 which is an almost integer. h(16) = 5315654681981354.5130767434 which is not an almost integer. h(17) = 130370767029135900.4579853491 which is not an almost integer.
Crystal
<lang ruby>require "big"
LN2 = Math.log(2).to_big_f
FACTORIALS = Hash(Int32, Float64).new{|h,k| h[k] = k * h[k-1]} FACTORIALS[0] = 1
def hickerson(n)
FACTORIALS[n] / (2 * LN2 ** (n+1))
end
def nearly_int?(n)
int = n.round (int - 0.1..int + 0.1).includes? n
end
1.upto(17) do |n|
h = hickerson(n) str = nearly_int?(h) ? "nearly integer" : "NOT nearly integer" puts "n:%3i h: %s\t%s" % [n, h, str]
end </lang>
- Output:
n: 1 h: 1.04068449050280396857 nearly integer n: 2 h: 3.00278070715690574485 nearly integer n: 3 h: 12.9962905052769682015 nearly integer n: 4 h: 74.9987354476616126737 nearly integer n: 5 h: 541.00151851642361617 nearly integer n: 6 h: 4683.00124726225853393 nearly integer n: 7 h: 47292.998731314636563 nearly integer n: 8 h: 545834.997907485331424 nearly integer n: 9 h: 7087261.00162290149215 nearly integer n: 10 h: 102247563.005271079641 nearly integer n: 11 h: 1622632572.99755070131 nearly integer n: 12 h: 28091567594.9815846588 nearly integer n: 13 h: 526858348381.001495064 nearly integer n: 14 h: 10641342970443.0898723 nearly integer n: 15 h: 230283190977853.160709 NOT nearly integer n: 16 h: 5315654681981357.53644 NOT nearly integer n: 17 h: 130370767029135978.97 NOT nearly integer
D
The D real type has enough precision for this task.
<lang d>void main() {
import std.stdio, std.algorithm, std.mathspecial;
foreach (immutable n; 1 .. 18) {
immutable x = gamma(n + 1) / (2 * LN2 ^^ (n + 1)),
tenths = cast(int)floor((x - x.floor) * 10);
writefln("H(%2d)=%22.2f is %snearly integer.", n, x,
tenths.among!(0, 9) ? "" : "NOT ");
}
}</lang>
H( 1)= 1.04 is nearly integer. H( 2)= 3.00 is nearly integer. H( 3)= 13.00 is nearly integer. H( 4)= 75.00 is nearly integer. H( 5)= 541.00 is nearly integer. H( 6)= 4683.00 is nearly integer. H( 7)= 47293.00 is nearly integer. H( 8)= 545835.00 is nearly integer. H( 9)= 7087261.00 is nearly integer. H(10)= 102247563.01 is nearly integer. H(11)= 1622632573.00 is nearly integer. H(12)= 28091567594.98 is nearly integer. H(13)= 526858348381.00 is nearly integer. H(14)= 10641342970443.08 is nearly integer. H(15)= 230283190977853.04 is nearly integer. H(16)= 5315654681981354.51 is NOT nearly integer. H(17)= 130370767029135900.50 is NOT nearly integer.
Factor
<lang factor>USING: formatting kernel math math.factorials math.functions math.ranges sequences ; IN: rosetta-code.hickerson
- ln2 ( -- x )
99 [1,b] [ [ 2 swap ^ ] [ * ] bi recip ] map-sum ;
- hickerson ( n -- x ) [ n! ] [ 1 + ln2 swap ^ 2 * ] bi / ;
- almost-int? ( x -- ? ) 10 * truncate 10 mod { 0 9 } member? ;
- hickerson-demo ( -- )
18 [1,b] [
dup hickerson dup almost-int?
"h(%02d) = %23.3f almost integer? %u\n" printf
] each ;
MAIN: hickerson-demo</lang>
- Output:
h(01) = 1.041 almost integer? t h(02) = 3.003 almost integer? t h(03) = 12.996 almost integer? t h(04) = 74.999 almost integer? t h(05) = 541.002 almost integer? t h(06) = 4683.001 almost integer? t h(07) = 47292.999 almost integer? t h(08) = 545834.998 almost integer? t h(09) = 7087261.002 almost integer? t h(10) = 102247563.005 almost integer? t h(11) = 1622632572.998 almost integer? t h(12) = 28091567594.982 almost integer? t h(13) = 526858348381.001 almost integer? t h(14) = 10641342970443.085 almost integer? t h(15) = 230283190977853.037 almost integer? t h(16) = 5315654681981354.513 almost integer? f h(17) = 130370767029135900.458 almost integer? f h(18) = 3385534663256845326.390 almost integer? f
Fortran
<lang fortran>program hickerson
implicit none integer, parameter :: q = selected_real_kind(30) integer, parameter :: l = selected_int_kind(15) real(q) :: s, l2 integer :: i, n, k
l2 = log(2.0_q)
do n = 1, 17
s = 0.5_q / l2
do i = 1, n
s = (s * i) / l2
end do
k = floor((s - floor(s, l)) * 10)
if (k == 0 .or. k == 9) then
print 1, n, s, ""
else
print 1, n, s, " NOT"
endif
end do
1 format('h(',I2,') = ',F23.3,' is',A,' an almost-integer')
end program</lang>
- Output:
h( 1) = 1.041 is an almost-integer h( 2) = 3.003 is an almost-integer h( 3) = 12.996 is an almost-integer h( 4) = 74.999 is an almost-integer h( 5) = 541.002 is an almost-integer h( 6) = 4683.001 is an almost-integer h( 7) = 47292.999 is an almost-integer h( 8) = 545834.998 is an almost-integer h( 9) = 7087261.002 is an almost-integer h(10) = 102247563.005 is an almost-integer h(11) = 1622632572.998 is an almost-integer h(12) = 28091567594.982 is an almost-integer h(13) = 526858348381.001 is an almost-integer h(14) = 10641342970443.085 is an almost-integer h(15) = 230283190977853.037 is an almost-integer h(16) = 5315654681981354.513 is NOT an almost-integer h(17) = 130370767029135900.458 is NOT an almost-integer
FreeBASIC
<lang freebasic>' version 08-10-2016 ' compile with: fbc -s gui
- Include Once "gmp.bi"
- Macro init_float_size (big_float ,size)
Dim As Mpf_ptr big_float = Allocate( Len( __mpf_struct)) Mpf_init2( big_float,size)
- EndMacro
- Macro mpf_remove(big_float)
Mpf_clear(big_float) DeAllocate (big_float)
- EndMacro
Screen 20
init_float_size(tmp,1024) init_float_size(ln2,1024) init_float_size(pow_ln2,2048) init_float_size(answer,2048)
Dim As UInteger n, num Dim As String st Dim As ZString Ptr text text = Allocate (1500) ' size 1500 char.
' need to calculate ln(2)
' x 1 1 1 1 ' ln --- = - + ------ + ------ ... + ------ ' x-1 x 2x^2 3x^3 nx^n
Mpf_set_ui(answer, 1) Mpf_div_ui(answer, answer, 2) ' answer = 1/2
For n = 2 To 1024
Mpf_set_ui(tmp, 2) ' tmp = x = 2 Mpf_pow_ui(tmp, tmp, n) ' tmp = x ^ n Mpf_mul_ui(tmp, tmp, n) ' tmp = n * (x ^ n) Mpf_ui_div(tmp, 1, tmp) ' tmp = 1 / (n * (x ^ n)) Mpf_add(answer, answer, tmp) ' answer += tmp
Next
/' remove this line if you want to print ln(2) Gmp_sprintf(text,!"ln(2) =\t %2.100Ff ", answer) Print *text Print '/
Mpf_set_ui(tmp, 1) Mpf_set(ln2, answer ) Mpf_set(pow_ln2, ln2)
Print For n = 1 To 40
Mpf_mul_ui(tmp, tmp, n) ' n!
Mpf_mul(pow_ln2, pow_ln2, ln2) ' ln(2)^(n+1)
Mpf_div_ui(answer, tmp , 2) ' / 2
Mpf_div(answer, answer, pow_ln2) ' / ln(2)^(n+1)
Gmp_sprintf(text,!"h(%i) =\t %65.5Ff ",n, answer)
st = *text
num = st[InStr(st,".")] ' get the first character after the "."
If num = Asc("0") Or num = Asc("9") Then
Color 10
Print st;" is a almost integer"
Color 15
Else
Color 12
Print st;" not close enough"
Color 15
End If
Next
mpf_remove(answer) 'cleanup mpf_remove(pow_ln2) mpf_remove(ln2) mpf_remove(tmp) DeAllocate (text)
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End</lang>
- Output:
h(1) = 1.04068 is a almost integer h(2) = 3.00278 is a almost integer h(3) = 12.99629 is a almost integer h(4) = 74.99874 is a almost integer h(5) = 541.00152 is a almost integer h(6) = 4683.00125 is a almost integer h(7) = 47292.99873 is a almost integer h(8) = 545834.99791 is a almost integer h(9) = 7087261.00162 is a almost integer h(10) = 102247563.00527 is a almost integer h(11) = 1622632572.99755 is a almost integer h(12) = 28091567594.98157 is a almost integer h(13) = 526858348381.00125 is a almost integer h(14) = 10641342970443.08453 is a almost integer h(15) = 230283190977853.03744 is a almost integer h(16) = 5315654681981354.51308 not close enough h(17) = 130370767029135900.45799 not close enough h(18) = 3385534663256845326.39030 not close enough h(19) = 92801587319328411139.87379 not close enough h(20) = 2677687796244384203087.52849 not close enough h(21) = 81124824998504073881728.73642 not close enough h(22) = 2574844419803190384544450.20255 not close enough h(23) = 85438451336745709294581778.58838 not close enough h(24) = 2958279121074145472650646597.12769 not close enough h(25) = 106697365438475775825583475660.43285 not close enough h(26) = 4002225759844168492486127555858.62763 not close enough h(27) = 155897763918621623249276226664346.97380 is a almost integer h(28) = 6297562064950066033518373935416160.57852 not close enough h(29) = 263478385263023690020893329036314027.57946 not close enough h(30) = 11403568794011880483742464196174527074.25258 not close enough h(31) = 510008036574269388430841024076099269561.72065 not close enough h(32) = 23545154085734896649184490637145314147689.71316 not close enough h(33) = 1120959742203056268267494209293002620410998.73593 not close enough h(34) = 54984904077825684862426868390301031843623475.17448 not close enough h(35) = 2776425695289206002630310219593685601350868792.44731 not close enough h(36) = 144199280951655469628360978109406918279521398522.98537 is a almost integer h(37) = 7697316738562185268347644943000493477791292787264.22720 not close enough h(38) = 421985466101260424678587486718115935816310614908533.21973 not close enough h(39) = 23743057231588741419119534567705900419845239863116450.98123 is a almost integer h(40) = 1370159636942236704917645663312384364387429049720377464.47614 not close enough
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
Go
<lang go>package main
import (
"fmt" "math/big"
)
func main() {
ln2, _ := new(big.Rat).SetString("0.6931471805599453094172")
h := big.NewRat(1, 2)
h.Quo(h, ln2)
var f big.Rat
var w big.Int
for i := int64(1); i <= 17; i++ {
h.Quo(h.Mul(h, f.SetInt64(i)), ln2)
w.Quo(h.Num(), h.Denom())
f.Sub(h, f.SetInt(&w))
y, _ := f.Float64()
d := fmt.Sprintf("%.3f", y)
fmt.Printf("n: %2d h: %18d%s Nearly integer: %t\n",
i, &w, d[1:], d[2] == '0' || d[2] == '9')
}
}</lang>
- Output:
n: 1 h: 1.041 Nearly integer: true n: 2 h: 3.003 Nearly integer: true n: 3 h: 12.996 Nearly integer: true n: 4 h: 74.999 Nearly integer: true n: 5 h: 541.002 Nearly integer: true n: 6 h: 4683.001 Nearly integer: true n: 7 h: 47292.999 Nearly integer: true n: 8 h: 545834.998 Nearly integer: true n: 9 h: 7087261.002 Nearly integer: true n: 10 h: 102247563.005 Nearly integer: true n: 11 h: 1622632572.998 Nearly integer: true n: 12 h: 28091567594.982 Nearly integer: true n: 13 h: 526858348381.001 Nearly integer: true n: 14 h: 10641342970443.085 Nearly integer: true n: 15 h: 230283190977853.037 Nearly integer: true n: 16 h: 5315654681981354.513 Nearly integer: false n: 17 h: 130370767029135900.458 Nearly integer: false
Haskell
<lang haskell>import Data.Number.CReal -- from numbers
import qualified Data.Number.CReal as C
hickerson :: Int -> C.CReal hickerson n = (fromIntegral $ product [1..n]) / (2 * (log 2 ^ (n + 1)))
charAfter :: Char -> String -> Char charAfter ch string = ( dropWhile (/= ch) string ) !! 1
isAlmostInteger :: C.CReal -> Bool isAlmostInteger = (`elem` ['0', '9']) . charAfter '.' . show
checkHickerson :: Int -> String checkHickerson n = show $ (n, hickerson n, isAlmostInteger $ hickerson n)
main :: IO () main = mapM_ putStrLn $ map checkHickerson [1..18]
</lang>
- Output:
(1,1.040684490502803898934790801867495712532,True) (2,3.0027807071569054434997674072193048895102,True) (3,12.9962905052769664622248845429643076304964,True) (4,74.9987354476616001276345503756447016691321,True) (5,541.0015185164235075692027746182564709468146,True) (6,4683.0012472622574371804671524790095859510537,True) (7,47292.9987313146239048228354877863088058057962,True) (8,545834.9979074851670672910397944923634653083422,True) (9,7087261.0016228991209791875159082204876452222335,True) (10,102247563.0052710420110883885941994702319297523418,True) (11,1622632572.9975500498528748610784990790541604568286,True) (12,28091567594.9815724407151891760995257984907158464557,True) (13,526858348381.0012482861804893808362085163957120038669,True) (14,10641342970443.0845319270950701503926391962048486244252,True) (15,230283190977853.0374360391259771068310386307484746008223,True) (16,5315654681981354.5130767434568055779953901730514005313021,False) (17,130370767029135900.4579853491967736302239049400128248109361,False) (18,3385534663256845326.3903018015167499502289072143201827212596,False)
J
Using to approximate the natural log of 2 (defining a function whose argument is the limit value for k).
Definitions:
<lang J>ln2=: [: +/ 1 % [: (*2x&^) 1+i. h=: ! % 2*(ln2 99)^>:</lang>
Implementation (1 in initial column indicates an almost integer, results displayed to 3 digits after the decimal point):
<lang J> 1 23j3": (h ,.~ 0 9 e.~ 10 <.@* 1 | h) x:1+i.17 1 1.041 1 3.003 1 12.996 1 74.999 1 541.002 1 4683.001 1 47292.999 1 545834.998 1 7087261.002 1 102247563.005 1 1622632572.998 1 28091567594.982 1 526858348381.001 1 10641342970443.085 1 230283190977853.037 0 5315654681981354.513 0 130370767029135900.458</lang>
In other words, multiply the fractional part of h by ten, find the largest integer not greater than this result, and check if it's in the set {0,9}. Then format the result of h along with this set membership result.
Java
<lang java>import java.math.*;
public class Hickerson {
final static String LN2 = "0.693147180559945309417232121458";
public static void main(String[] args) {
for (int n = 1; n <= 17; n++)
System.out.printf("%2s is almost integer: %s%n", n, almostInteger(n));
}
static boolean almostInteger(int n) {
BigDecimal a = new BigDecimal(LN2);
a = a.pow(n + 1).multiply(BigDecimal.valueOf(2));
long f = n;
while (--n > 1)
f *= n;
BigDecimal b = new BigDecimal(f);
b = b.divide(a, MathContext.DECIMAL128);
BigInteger c = b.movePointRight(1).toBigInteger().mod(BigInteger.TEN);
return c.toString().matches("0|9");
}
}</lang>
1 is almost integer: true 2 is almost integer: true 3 is almost integer: true 4 is almost integer: true 5 is almost integer: true 6 is almost integer: true 7 is almost integer: true 8 is almost integer: true 9 is almost integer: true 10 is almost integer: true 11 is almost integer: true 12 is almost integer: true 13 is almost integer: true 14 is almost integer: true 15 is almost integer: true 16 is almost integer: false 17 is almost integer: false
jq
jq currently uses IEEE 754 64-bit numbers, and therefore the built-in arithmetic functions lack adequate precision to solve the task completely. In the following, therefore, we include a check for adequate precision. <lang jq>def hickerson:
. as $n | (2|log) as $log2 | reduce range(1;$n+1) as $i ( 0.5/$log2; . * $i / $log2) ;
def precise:
(. - 0.05) as $x | . != ($x + 0.1) ;
def almost_an_integer:
tostring
| index(".") as $ix
| if $ix == null then true
else .[$ix+1:$ix+2] | (. == "9" or . == "0")
end ;
range(1;18)
| . as $i
| hickerson
| if precise then
if almost_an_integer
then "hickerson(\($i)) is \(.) -- almost an integer"
else "hickerson(\($i)) is \(.)"
end
else "insufficient precision for hickerson(\($i))"
end</lang>
- Output:
<lang sh>$ jq -M -r -n -f Hickerson_series.jq hickerson(1) is 1.0406844905028039 -- almost an integer hickerson(2) is 3.0027807071569055 -- almost an integer hickerson(3) is 12.996290505276967 -- almost an integer hickerson(4) is 74.9987354476616 -- almost an integer hickerson(5) is 541.0015185164235 -- almost an integer hickerson(6) is 4683.001247262258 -- almost an integer hickerson(7) is 47292.99873131463 -- almost an integer hickerson(8) is 545834.9979074852 -- almost an integer hickerson(9) is 7087261.001622899 -- almost an integer hickerson(10) is 102247563.00527105 -- almost an integer hickerson(11) is 1622632572.9975502 -- almost an integer hickerson(12) is 28091567594.98158 -- almost an integer hickerson(13) is 526858348381.00134 -- almost an integer hickerson(14) is 10641342970443.086 -- almost an integer hickerson(15) is 230283190977853.06 -- almost an integer insufficient precision for hickerson(16) insufficient precision for hickerson(17)</lang>
Julia
This solution implements its Hickerson Series function as a closure. It explores the effects of datatype precision, implementing a rather conservative "guard number" scheme to identify when the results may have inadequate precision. One can not be confident of the 64-bit floating point results beyond n=13, but the 256-bit precision floating point results are easily precise enough for the entire calculation to n=17. (A slight variant of this calculation, not shown here, indicates that these extended precision numbers are adequate to n=50.) <lang Julia> function makehickerson{T<:Real}(x::T)
n = 0
h = one(T)/2x
function hickerson()
n += 1
h *= n/x
end
end
function reporthickerson{T<:Real,U<:Integer}(a::T, nmax::U)
h = makehickerson(a) hgm = makehickerson(prevfloat(a)) hgp = makehickerson(nextfloat(a))
println()
print("Performing calculations using ", typeof(a))
println(", which has ", precision(a), "-bit precision.")
for i in 1:nmax
x = h()
xm = hgm()
xp = hgp()
y = ifloor(10x)
ym = ifloor(10xm)
yp = ifloor(10xp)
println()
println("Hickerson series result for n = ", i)
println(@sprintf(" -> %25.4f ", xm))
println(@sprintf(" 0> %25.4f ", x))
println(@sprintf(" +> %25.4f ", xp))
isprecok =
isint =
if ym == y == yp
print("The precision is adequate, ")
if digits(y)[1] in [0, 9]
println("and the result is an almost integer.")
else
println("but the result is not an almost integer.")
end
else
println("The precision is inadequate for a definite result.")
end
end
end
a = log(big(2.0)) reporthickerson(a, 17)
a = log(2.0) reporthickerson(a, 17) </lang>
- Output:
Performing calculations using BigFloat, which has 256-bit precision.
Hickerson series result for n = 1
-> 1.0407
0> 1.0407
+> 1.0407
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 2
-> 3.0028
0> 3.0028
+> 3.0028
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 3
-> 12.9963
0> 12.9963
+> 12.9963
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 4
-> 74.9987
0> 74.9987
+> 74.9987
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 5
-> 541.0015
0> 541.0015
+> 541.0015
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 6
-> 4683.0012
0> 4683.0012
+> 4683.0012
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 7
-> 47292.9987
0> 47292.9987
+> 47292.9987
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 8
-> 545834.9979
0> 545834.9979
+> 545834.9979
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 9
-> 7087261.0016
0> 7087261.0016
+> 7087261.0016
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 10
-> 102247563.0053
0> 102247563.0053
+> 102247563.0053
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 11
-> 1622632572.9976
0> 1622632572.9976
+> 1622632572.9976
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 12
-> 28091567594.9816
0> 28091567594.9816
+> 28091567594.9816
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 13
-> 526858348381.0012
0> 526858348381.0012
+> 526858348381.0012
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 14
-> 10641342970443.0845
0> 10641342970443.0845
+> 10641342970443.0845
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 15
-> 230283190977853.0374
0> 230283190977853.0374
+> 230283190977853.0374
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 16
-> 5315654681981354.5131
0> 5315654681981354.5131
+> 5315654681981354.5131
The precision is adequate, but the result is not an almost integer.
Hickerson series result for n = 17
-> 130370767029135900.4580
0> 130370767029135900.4580
+> 130370767029135900.4580
The precision is adequate, but the result is not an almost integer.
Performing calculations using Float64, which has 53-bit precision.
Hickerson series result for n = 1
-> 1.0407
0> 1.0407
+> 1.0407
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 2
-> 3.0028
0> 3.0028
+> 3.0028
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 3
-> 12.9963
0> 12.9963
+> 12.9963
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 4
-> 74.9987
0> 74.9987
+> 74.9987
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 5
-> 541.0015
0> 541.0015
+> 541.0015
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 6
-> 4683.0012
0> 4683.0012
+> 4683.0012
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 7
-> 47292.9987
0> 47292.9987
+> 47292.9987
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 8
-> 545834.9979
0> 545834.9979
+> 545834.9979
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 9
-> 7087261.0016
0> 7087261.0016
+> 7087261.0016
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 10
-> 102247563.0053
0> 102247563.0053
+> 102247563.0053
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 11
-> 1622632572.9976
0> 1622632572.9976
+> 1622632572.9975
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 12
-> 28091567594.9816
0> 28091567594.9816
+> 28091567594.9815
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 13
-> 526858348381.0024
0> 526858348381.0015
+> 526858348381.0003
The precision is adequate, and the result is an almost integer.
Hickerson series result for n = 14
-> 10641342970443.1094
0> 10641342970443.0898
+> 10641342970443.0645
The precision is inadequate for a definite result.
Hickerson series result for n = 15
-> 230283190977853.5938
0> 230283190977853.1563
+> 230283190977852.5625
The precision is inadequate for a definite result.
Hickerson series result for n = 16
-> 5315654681981368.0000
0> 5315654681981357.0000
+> 5315654681981343.0000
The precision is inadequate for a definite result.
Hickerson series result for n = 17
-> 130370767029136256.0000
0> 130370767029135968.0000
+> 130370767029135600.0000
The precision is inadequate for a definite result.
Kotlin
<lang kotlin>// version 1.1.4
import java.math.BigDecimal import java.math.BigInteger import java.math.MathContext
object Hickerson {
private const val LN2 = "0.693147180559945309417232121458"
fun almostInteger(n: Int): Boolean {
val a = BigDecimal(LN2).pow(n + 1) * BigDecimal(2)
var nn = n
var f = n.toLong()
while (--nn > 1) f *= nn
val b = BigDecimal(f).divide(a, MathContext.DECIMAL128)
val c = b.movePointRight(1).toBigInteger() % BigInteger.TEN
return c.toString().matches(Regex("[09]"))
}
}
fun main(args: Array<String>) {
for (n in 1..17) println("${"%2d".format(n)} is almost integer: ${Hickerson.almostInteger(n)}")
}</lang>
- Output:
1 is almost integer: true 2 is almost integer: true 3 is almost integer: true 4 is almost integer: true 5 is almost integer: true 6 is almost integer: true 7 is almost integer: true 8 is almost integer: true 9 is almost integer: true 10 is almost integer: true 11 is almost integer: true 12 is almost integer: true 13 is almost integer: true 14 is almost integer: true 15 is almost integer: true 16 is almost integer: false 17 is almost integer: false
Mathematica / Wolfram Language
<lang Mathematica>h[n_] = n!/(2 (Log[2])^(n + 1)); firstdecimal[x_] := Floor[10 x] - 10 Floor[x]; almostIntegerQ[x_] := firstdecimal[x] == 0 || firstdecimal[x] == 9; Table[{i, AccountingForm[N[h[i], 50], {Infinity, 5}], almostIntegerQ[N[h[i], 50]]}, {i, 1, 17}] // TableForm</lang>
- Output:
1 1.04068 True 2 3.00278 True 3 12.99629 True 4 74.99874 True 5 541.00152 True 6 4683.00125 True 7 47292.99873 True 8 545834.99791 True 9 7087261.00162 True 10 102247563.00527 True 11 1622632572.99755 True 12 28091567594.98157 True 13 526858348381.00125 True 14 10641342970443.08453 True 15 230283190977853.03744 True 16 5315654681981354.51308 False 17 130370767029135900.45799 False
ML
mLite
<lang ocaml>local
fun log (n, k, last = sum, sum) = sum
| (n, k, last, sum) = log (n, k + 1, sum, sum + ( 1 / (k * n ^ k)))
| n = log (n, 1, 1, 0);
val ln2 = log 2
in
fun hickerson n = (fold (opt *,1) ` iota n) / (2 * ln2 ^ (n+1));
fun almost_int ([]) = false
| (h :: h' :: t) where (h <> #".") = almost_int (h' :: t)
| (h :: h' :: t) = h' = #"0" or h' = #"9"
| s = almost_int (explode s)
and show (n, h) = println (n, ntos h, almost_int ` ntos ` h)
| n = show (n, hickerson n)
end
foreach show ` iota 17; </lang> Output
(1, 1.04068449050280389893479080186749587, true) (2, 3.00278070715690544349976740721930557, true) (3, 12.9962905052769664622248845429643116, true) (4, 74.9987354476616001276345503756447306, true) (5, 541.001518516423507569202774618256721, true) (6, 4683.0012472622574371804671524790121, true) (7, 47292.9987313146239048228354877863377, true) (8, 545834.997907485167067291039794492739, true) (9, 7087261.00162289912097918751590822596, true) (10, 102247563.005271042011088388594199557, true) (11, 1622632572.99755004985287486107850058, true) (12, 28091567594.9815724407151891760995539, true) (13, 526858348381.001248286180489380836777, true) (14, 10641342970443.0845319270950701504049, true) (15, 230283190977853.037436039125977107112, true) (16, 5315654681981354.51307674345680558492, false) (17, 130370767029135900.457985349196773809, false)
PARI/GP
<lang parigp>h(n)=n!/2/log(2)^(n+1) almost(x)=abs(x-round(x))<.1 select(n->almost(h(n)),[1..17]) </lang>
- Output:
%1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
Perl
We'll use Math::BigFloat, which comes with perl.
Conveniently, it includes a sufficiently precise logarithm function.
To avoid doing factorials or exponentiations or divisions in the loop, we first calculate the inverse of the natural log of 2, and then compute a running value of h.
<lang perl>use strict; use warnings; use Math::BigFloat;
my $iln2 = 1 / Math::BigFloat->new(2)->blog; my $h = $iln2 / 2;
for my $n ( 1 .. 17 ) { $h *= $iln2; $h *= $n; my $s = $h->copy->bfround(-3)->bstr; printf "h(%2d) = %22s is%s almost an integer.\n", $n, $s, ($s =~ /\.[09]/ ? "" : " NOT"); } </lang>
- Output:
h( 1) = 1.041 is almost an integer. h( 2) = 3.003 is almost an integer. h( 3) = 12.996 is almost an integer. h( 4) = 74.999 is almost an integer. h( 5) = 541.002 is almost an integer. h( 6) = 4683.001 is almost an integer. h( 7) = 47292.999 is almost an integer. h( 8) = 545834.998 is almost an integer. h( 9) = 7087261.002 is almost an integer. h(10) = 102247563.005 is almost an integer. h(11) = 1622632572.998 is almost an integer. h(12) = 28091567594.982 is almost an integer. h(13) = 526858348381.001 is almost an integer. h(14) = 10641342970443.085 is almost an integer. h(15) = 230283190977853.037 is almost an integer. h(16) = 5315654681981354.513 is NOT almost an integer. h(17) = 130370767029135900.458 is NOT almost an integer.
Phix
<lang Phix>include bigatom.e bigatom ln2 = ba_log(2),
hn = ba_divide(0.5,ln2)
for n=1 to 17 do
hn = ba_divide(ba_multiply(n,hn),ln2)
string n10 = ba_sprintf("%24.4aB ",hn)
n10 &= iff(find(n10[21],"09")?"Y":"N")
printf(1,"%2d:%s\n",{n,n10})
end for</lang>
- Output:
1: 1.0407 Y 2: 3.0028 Y 3: 12.9963 Y 4: 74.9987 Y 5: 541.0015 Y 6: 4683.0012 Y 7: 47292.9987 Y 8: 545834.9979 Y 9: 7087261.0016 Y 10: 102247563.0053 Y 11: 1622632572.9976 Y 12: 28091567594.9816 Y 13: 526858348381.0012 Y 14: 10641342970443.0845 Y 15: 230283190977853.0374 Y 16: 5315654681981354.5131 N 17: 130370767029135900.4580 N
PicoLisp
PicoLisp does not have floating point, but does have library functions to make fixed-point math easier. I use a scale factor of 25 to ensure that there is enough precision. <lang PicoLisp> (load "@lib/misc.l")
(scl 25)
(load "@lib/misc.l")
(scl 25)
(setq LN2 0.69314718055994530941723212)
(de almost-int? (N)
(bool (member (% (/ N 0.1) 10) (0 9))))
(de fmt4 (N)
(format (/ N 0.0001) 4))
(de h (N)
(*/ (factorial N) `(* 1.0 1.0) (* 2 (s** LN2 (inc N)))))
(de s** (A N) # scaled integer exponentiation
(let R 1.0
(for Bit (chop (bin N))
(setq R (*/ R R 1.0))
(when (= Bit "1")
(setq R (*/ A R 1.0))))
R))
(de factorial (N)
(cond
((lt0 N) NIL)
((=0 N) 1)
(T (apply * (range 1 N)))))
(for I 18
(let H (h I)
(if (almost-int? H)
(prin "yes: ")
(prin " no: "))
(prinl "h(" (align 2 I) ") = " (fmt4 H))))
(bye) </lang>
- Output:
yes: h( 1) = 1.0406 yes: h( 2) = 3.0027 yes: h( 3) = 12.9962 yes: h( 4) = 74.9987 yes: h( 5) = 541.0015 yes: h( 6) = 4683.0012 yes: h( 7) = 47292.9987 yes: h( 8) = 545834.9979 yes: h( 9) = 7087261.0016 yes: h(10) = 102247563.0052 yes: h(11) = 1622632572.9975 yes: h(12) = 28091567594.9815 yes: h(13) = 526858348381.0012 yes: h(14) = 10641342970443.0845 yes: h(15) = 230283190977853.0374 no: h(16) = 5315654681981354.5130 no: h(17) = 130370767029135900.4579 no: h(18) = 3385534663256845326.3904
PL/I
<lang pli>Hickerson: procedure options (main); /* 12 January 2014 */
declare s float (18), (i, n) fixed binary; declare is fixed decimal (30); declare f fixed decimal (4,3);
do n = 1 to 16; /* 17 exceeds hardware precision */
s = 0.5Q0 / 0.693147180559945309417232121458;
do i = 1 to n;
s = (s * i) / 0.693147180559945309417232121458;
end;
is = s;
f = s-is;
is = 10*f;
put skip edit (n, s) (f(2), f(30,3));
if is = 0 | is = 9 then
put edit (' is a near integer') (A);
else
put edit (' is not a near integer') (A);
end;
end Hickerson;</lang>
- Output:
1 1.041 is a near integer 2 3.003 is a near integer 3 12.996 is a near integer 4 74.999 is a near integer 5 541.002 is a near integer 6 4683.001 is a near integer 7 47292.999 is a near integer 8 545834.998 is a near integer 9 7087261.002 is a near integer 10 102247563.005 is a near integer 11 1622632572.998 is a near integer 12 28091567594.982 is a near integer 13 526858348381.001 is a near integer 14 10641342970443.085 is a near integer 15 230283190977853.037 is a near integer 16 5315654681981354.510 is not a near integer
Using extended precision available by software: <lang pli> do n = 1 to 18;
s = divide( float(0.5), float(0.693147180559945309417232121458) );
do i = 1 to n;
s = divide ( multiply(s, float(i)), float(0.693147180559945309417232121458) );
end;
is = trunc(s);
f = subtract(s, is);
is = multiply(float(10), f);
is = trunc(is);
put skip edit (n) (f(3)); call output_fixed (s, 'F(60,4)' );
if compare(is, float(0)) = 0 | compare(is, float(9)) = 0 then
put edit (' is a near integer') (A);
else
put edit (' is not a near integer') (A);
end;
</lang>
- Output:
Results
1 1.0406 is a near integer 2 3.0027 is a near integer 3 12.9962 is a near integer 4 74.9987 is a near integer 5 541.0015 is a near integer 6 4683.0012 is a near integer 7 47292.9987 is a near integer 8 545834.9979 is a near integer 9 7087261.0016 is a near integer 10 102247563.0052 is a near integer 11 1622632572.9975 is a near integer 12 28091567594.9815 is a near integer 13 526858348381.0012 is a near integer 14 10641342970443.0845 is a near integer 15 230283190977853.0374 is a near integer 16 5315654681981354.5130 is not a near integer 17 130370767029135900.4579 is not a near integer 18 3385534663256845326.3903 is not a near integer
Python
This uses Pythons decimal module of fixed precision decimal floating point calculations.
<lang python>from decimal import Decimal import math
def h(n):
'Simple, reduced precision calculation' return math.factorial(n) / (2 * math.log(2) ** (n + 1))
def h2(n):
'Extended precision Hickerson function' return Decimal(math.factorial(n)) / (2 * Decimal(2).ln() ** (n + 1))
for n in range(18):
x = h2(n)
norm = str(x.normalize())
almostinteger = (' Nearly integer'
if 'E' not in norm and ('.0' in norm or '.9' in norm)
else ' NOT nearly integer!')
print('n:%2i h:%s%s' % (n, norm, almostinteger))</lang>
- Output:
n: 0 h:0.7213475204444817036799623405 NOT nearly integer! n: 1 h:1.040684490502803898934790802 Nearly integer n: 2 h:3.002780707156905443499767406 Nearly integer n: 3 h:12.99629050527696646222488454 Nearly integer n: 4 h:74.99873544766160012763455035 Nearly integer n: 5 h:541.0015185164235075692027746 Nearly integer n: 6 h:4683.001247262257437180467151 Nearly integer n: 7 h:47292.99873131462390482283547 Nearly integer n: 8 h:545834.9979074851670672910395 Nearly integer n: 9 h:7087261.001622899120979187513 Nearly integer n:10 h:102247563.0052710420110883885 Nearly integer n:11 h:1622632572.997550049852874859 Nearly integer n:12 h:28091567594.98157244071518915 Nearly integer n:13 h:526858348381.0012482861804887 Nearly integer n:14 h:10641342970443.08453192709506 Nearly integer n:15 h:230283190977853.0374360391257 Nearly integer n:16 h:5315654681981354.513076743451 NOT nearly integer! n:17 h:130370767029135900.4579853491 NOT nearly integer!
The range for should be reduced to be for this definition of almost integer.
Racket
<lang Racket>#lang racket (require math/bigfloat)
(define (hickerson n)
(bf/ (bffactorial n)
2.bf
(bfexpt (bflog 2.bf) (bf (+ n 1)))))
(for ([n (in-range 18)])
(define hickerson-n (hickerson n))
(define first-decimal
(bigfloat->integer (bftruncate (bf* 10.bf (bffrac hickerson-n)))))
(define almost-integer (if (or (= first-decimal 0) (= first-decimal 9))
"is Nearly integer"
"is not Nearly integer!"))
(printf "~a: ~a ~a\n"
(~r n #:min-width 2)
(bigfloat->string hickerson-n)
almost-integer))</lang>
- Output:
0: 0.7213475204444817036799623405009460687136 is not Nearly integer! 1: 1.040684490502803898934790801867495712535 is Nearly integer 2: 3.002780707156905443499767407219304889517 is Nearly integer 3: 12.99629050527696646222488454296430763056 is Nearly integer 4: 74.99873544766160012763455037564470166964 is Nearly integer 5: 541.0015185164235075692027746182564709501 is Nearly integer 6: 4683.001247262257437180467152479009585976 is Nearly integer 7: 47292.99873131462390482283548778630880614 is Nearly integer 8: 545834.9979074851670672910397944923634685 is Nearly integer 9: 7087261.001622899120979187515908220487726 is Nearly integer 10: 102247563.005271042011088388594199470233 is Nearly integer 11: 1622632572.997550049852874861078499079071 is Nearly integer 12: 28091567594.98157244071518917609952579893 is Nearly integer 13: 526858348381.0012482861804893808362085239 is Nearly integer 14: 10641342970443.08453192709507015039263936 is Nearly integer 15: 230283190977853.0374360391259771068310425 is Nearly integer 16: 5315654681981354.513076743456805577995494 is not Nearly integer! 17: 130370767029135900.4579853491967736302263 is not Nearly integer!
Raku
(formerly Perl 6)
We'll use FatRat values, and a series for an approximation of ln(2).
<lang perl6>constant ln2 = [+] (1/2.FatRat, */2 ... *) Z/ 1 .. 100; constant h = [\*] 1/2, |(1..*) X/ ln2;
use Test; plan *;
for h[1..17] {
ok m/'.'<[09]>/, .round(0.001)
}</lang>
- Output:
ok 1 - 1.041 ok 2 - 3.003 ok 3 - 12.996 ok 4 - 74.999 ok 5 - 541.002 ok 6 - 4683.001 ok 7 - 47292.999 ok 8 - 545834.998 ok 9 - 7087261.002 ok 10 - 102247563.005 ok 11 - 1622632572.998 ok 12 - 28091567594.982 ok 13 - 526858348381.001 ok 14 - 10641342970443.085 ok 15 - 230283190977853.037 not ok 16 - 5315654681981354.513 # Failed test '5315654681981354.513' # at hickerson line 8 not ok 17 - 130370767029135900.458 # Failed test '130370767029135900.458' # at hickerson line 8
REXX
version 1
<lang rexx>/* REXX ---------------------------------------------------------------
- 04.01.2014 Walter Pachl - using a rather aged ln function of mine
- with probably unreasonably high precision
- 05.01.2014 added n=18
- --------------------------------------------------------------------*/
Numeric Digits 100 Do n=1 To 18
x=format(def(),20,10) Parse Var x '.' +1 d +1 If pos(d,'09')>0 Then tag='almost an integer' Else tag= Say right(n,2) x tag End
Exit
def:
x=fact(n)/(2*ln(2,200)**(n + 1)) Return x
ln: Procedure /***********************************************************************
- Return ln(x) -- with specified precision
- Three different series are used for the ranges 0 to 0.5
- 0.5 to 1.5
- 1.5 to infinity
- 920903 Walter Pachl
- /
Parse Arg x,prec,b
If prec= Then prec=9
Numeric Digits (2*prec)
Numeric Fuzz 3
Select
When x<=0 Then r='*** invalid argument ***'
When x<0.5 Then Do
z=(x-1)/(x+1)
o=z
r=z
k=1
Do i=3 By 2
ra=r
k=k+1
o=o*z*z
r=r+o/i
If r=ra Then Leave
End
r=2*r
End
When x<1.5 Then Do
z=(x-1)
o=z
r=z
k=1
Do i=2 By 1
ra=r
k=k+1
o=-o*z
r=r+o/i
If r=ra Then Leave
End
End
Otherwise /* 1.5<=x */ Do
z=(x+1)/(x-1)
o=1/z
r=o
k=1
Do i=3 By 2
ra=r
k=k+1
o=o/(z*z)
r=r+o/i
If r=ra Then Leave
End
r=2*r
End
End
If b<> Then
r=r/ln(b)
Numeric Digits (prec)
Return r+0
fact: Procedure
Parse Arg m fact=1 Do i=2 To m fact=fact*i End Return fact</lang>
- Output:
1 1.0406844905 almost an integer 2 3.0027807072 almost an integer 3 12.9962905053 almost an integer 4 74.9987354477 almost an integer 5 541.0015185164 almost an integer 6 4683.0012472623 almost an integer 7 47292.9987313146 almost an integer 8 545834.9979074852 almost an integer 9 7087261.0016228991 almost an integer 10 102247563.0052710420 almost an integer 11 1622632572.9975500499 almost an integer 12 28091567594.9815724407 almost an integer 13 526858348381.0012482862 almost an integer 14 10641342970443.0845319271 almost an integer 15 230283190977853.0374360391 almost an integer 16 5315654681981354.5130767435 17 130370767029135900.4579853492 18 3385534663256845326.3903018015
version 2
This REXX version can calculate the Hickerson series to any number up to the length (in decimal digits)
of the number returned by the ln2 function.
This version supports up to 507 decimal digits. <lang rexx>/*REXX program to calculate and show the Hickerson series (are near integer). */ numeric digits length(ln2())-length(.) /*be able to calculate big factorials. */ parse arg N . /*get optional number of values to use.*/ if N== then N= 1 /*Not specified? Then use the default.*/
/* [↓] compute possible Hickerson #s. */
do j=1 for N; #= Hickerson(j) /*traipse thru a range of Hickerson #s.*/
?= right(# * 10 % 1, 1) /*get 1st decimal digit past dec. point*/
if ?==0 | ?==9 then _= 'almost integer' /*the number is, */
else _= ' ' /* or it ain't. */
say right(j,3) _ format(#, , 5) /*show number with 5 dec digs fraction.*/
end /*j*/
exit /*stick a fork in it, we're all done. */ /*─────────────────────────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; parse arg x; !=1; do j=2 to x; != !*j; end; return ! /* ◄──compute X factorial*/ Hickerson: procedure; parse arg z; return !(z) / (2*ln2() ** (z+1)) ln2: return .69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641,
|| 8687542001481020570685733685520235758130557032670751635075961930727570828371435190307038623891673471123, || 3501153644979552391204751726815749320651555247341395258829504530070953263666426541042391578149520437404, || 3038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066, || 83411372738737229289564935470257626520988596932019650585547647033067936544325476327449512504060694381470</lang>
- output when using the input of: 197
1 almost integer 1.04068 2 almost integer 3.00278 3 almost integer 12.99629 4 almost integer 74.99874 5 almost integer 541.00152 6 almost integer 4683.00125 7 almost integer 47292.99873 8 almost integer 545834.99791 9 almost integer 7087261.00162 10 almost integer 102247563.00527 11 almost integer 1622632572.99755 12 almost integer 28091567594.98157 13 almost integer 526858348381.00125 14 almost integer 10641342970443.08453 15 almost integer 230283190977853.03744 16 5315654681981354.51308 17 130370767029135900.45799 18 3385534663256845326.39030 19 92801587319328411139.87379 20 2677687796244384203087.52849 21 81124824998504073881728.73642 22 2574844419803190384544450.20255 23 85438451336745709294581778.58838 24 2958279121074145472650646597.12769 25 106697365438475775825583475660.43285 26 4002225759844168492486127555858.62763 27 almost integer 155897763918621623249276226664346.97380 28 6297562064950066033518373935416160.57852 29 263478385263023690020893329036314027.57946 30 11403568794011880483742464196174527074.25258 31 510008036574269388430841024076099269561.72065 32 23545154085734896649184490637145314147689.71316 33 1120959742203056268267494209293002620410998.73593 34 54984904077825684862426868390301031843623475.17448 35 2776425695289206002630310219593685601350868792.44731 36 almost integer 144199280951655469628360978109406918279521398522.98537 37 7697316738562185268347644943000493477791292787264.22720 38 421985466101260424678587486718115935816310614908533.21973 39 almost integer 23743057231588741419119534567705900419845239863116450.98123 40 1370159636942236704917645663312384364387429049720377464.47614 41 81045623051154285047127402304207782853156218361814319021.28280 42 4910812975389574954318759599939388855544732309560436182821.35633 43 304646637632091740261982544696657582136519509662728787385223.78746 44 19338536506753895707368358095646384573117827333115778939657189.34970 45 1255482482235481041484313695469155949742941813368110258119125666.54828 46 almost integer 83318804148028351409201335290659562069258599819486505697819558094.95701 47 5649570401186486930330812460375430692673276471701214788305128308192.63131 48 391229145645351175841837029639030040330277058722445073547950709497937.24947 49 27656793065414932606012896651489726461435178241054470446176414701300759.68492 50 1995015910118319790635433747742913123711612309012803454341376306544687943.66054 51 almost integer 146788177561136377730260697283316629866051820967150940915169721150316606892.05918 52 11012069943086163504795579947992458193990796847590872523330083892852015565626.28941 53 842014110931079332783532220015836831045327371743757681552681009426224153091637.23798 54 65597557438735074090994025536102763729036609801731831230533885501204932903110195.70938 55 5205049894621061289218474196429496131424689228776492621372505509295323231187794811.18045 56 almost integer 420520781550767894712366209445415465306816295045705598150082071298498563284781170169.03701 57 almost integer 34580945029640504393370721771069401074083502173524617793574204974469998047025866870097.02237 58 2893605958403939796891998166457291344321113091137014546099293839148904090004806176427638.60317 59 246300867022091084352691958585695642066952827239759473981562041651101948387611615902530101.23216 60 almost integer 21320222365165370148450725694202652367764740960338726836059360731081589392256498745759003771.01903 61 1876273323689316794552898932003548830214301148023334732450711143880068675273459337334678356510.54479 62 almost integer 167827193605207470382244239593751346592925059206583127133257042316428312353166326463498634995723.96506 63 almost integer 15253777976254320482680028798302887920587781886722393040786662312902435640815721495160588622047541.92840 64 1408419189834457368564204829523128565178390518208834734403870767459047244122454099122442802088884321.73467 65 132074759743356507265849312929567147234908587643858117900265185019357132816098895359362429954543792250.41307 66 12575877659950554472502916897483150472428551212111494310419777827319052677957994891799291825191389852999.17201 67 1215591474434076764736685372169536683582142127785487987047068194303571231853163199814716027890404260071569.88314 68 almost integer 119253489850080307228410908979332327602349548167435756238732178160360682219616027578362147289795251646339953.92335 69 almost integer 11871202870663510214658796273851712937873144830499188934497638900478929836299976837994531733424834494413388915.95250 70 1198856785762515085433047378179373006408841522298485701569549736200218795168354301822880350705072744254029177974.27548 71 122800516508452069087008691134168306922157250579355968528220277637680521063341532176649945604606722169422881949772.27313 72 12755786125344990028594343963936309999106975215043630050197141450003929225895159286410920090232666677567781668116842.80093 73 1343397785154308762546986584839633417565562502594324180627238421955041065318679414963300088378553801208008677342105129.20781 74 143420385871167038557536299613944875171590207217392651473614091977309061787666287331580822225139240178311453147278916363.35957 75 15518390959403568033551566827678607140363326576559695820270383745035406196325951714541143408200761091941463352557779137479.59341 76 almost integer 1701511231657781450488490657187463365688823186902664849351256426751462495076884249622278025237121805817453644028642475582941.04704 77 189016659826575619359816463566590900567665644326734831940281352836144822599463416848350164874138510566588334766260387690390882.63636 78 21270085026621350415799651921499346803894882526119364175378674541895973741657242545930525335438800297104058164210172490673537333.32584 79 2424213448788264179583061306089147620134038095441867764757968971686974539641887227399315328126639570887068398905864468400203147379.13428 80 279792057649852783175737713059565268313828016234531102244960765250050772031108759182312138156720330707487174908389146019680769950627.13659 81 32696023738176487003017678437152831751829536269695148312421532512438766629331326025007830371276220306319964008365309960094718678123025.72933 82 3867972086916113699289650561448774985264647530291049879502944190830421004287744690833208017718554836505317767698155032416161235799978153.55849 83 almost integer 463165244291537377407938377908509856879151672553650449432252233056386498022670707383927182825531345754093757771505246691827175409837333706.03165 84 56129320888328204336159568944138094624862468930689936507266999153981195420801200053053466593736461319060626598320474363329095816269614248570.26351 85 6883086895994794564193512707959076687449938175712429198130223275797343967727966873924934477871130478678856374089795957959702559434133276875558.36346 86 853996798453916859286707332364040151020472833921624157042891999067692861031567447908099758601565668309504205967050579936439746813730932889524110.10646 87 almost integer 107188954307613016000523311392809231433160190250667607690375475681677727720687471319593137707247355623545849957670511278864612577019674851693529129.02379 88 13608405607954695161891393427685656221057735793458790720461284212774892489827559568850855692362627497225211166876726460186939885169175550464962373899.34048 89 1747317356365160010675286089449398905752478506010237186666200281047566557280027666701240190903491017547173702078796478817511283115112530023987934885247.58668 90 almost integer 226876147639850697035498164180665227340867432554801341517337271449480622375293088050301117860485783921944898315478899240634590962769072809212926135838226.00884 91 29785491471736445918952653271075531391540819876778975402174887327425599452642406068862719850771752593361583104778259003866002872441102216080600657543852651.65325 92 3953367036977750808685756007120090736939292055073874653365943059202128556342358488546018279923077047134125498022074379841398482263974194955368267319577754995.27286 93 almost integer 530425780772737757181826423434492823619343330766440565137249239877583887700661622746755246932052806628510661960680329286085407045591157790622109718215821384486.92752 94 71932808487165612459817313420088981223441387545508285691783916644870517242503482223932321736730312940138541874552901642390084976544951541683244641434588659192963.57883 95 almost integer 9858825077756689892365727469356584444697619626867085315546360659481108594876947783837934569215103038786902497849668159869688315692702653014667407310567738222438995.92792 96 1365434692672447254549334110811913069561967885901464921338743306353836651045642292387627910698830813843896593548335731977839012531770052413043446025832262165782270740.75950 97 191080868398299691145372728810441155882257482078053536497729039741289151762976230835088279174422983509793335031628712195215374587400810640276485642254628643079607175392.85664 98 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148283631065792687061419360565375379308559840321757580470062533908445626372097828715429287981460169451241740535123834324194616768608497984202142352223761497549980831803636384122583512842707655.33220 109 23318158451015424228950671773583457159746952971274726427350763676476539678958570327282116705779386714745323345804539112039982994738995010310011179032182763747601490575140661761177951120275275414.20366 110 almost integer 3700509071593732759743387879122534531093459460019546885341496988364005534089830811680294792747194445415434375962412398953950450394440991138741768593093575004323295113512001416462649321025727206777.03051 111 592596375585171933390750659287342641654953297691010994963323859733044942221570118622885921630262738836076821004543838157482529421781108680909554473521249711877254209263603384825050196032113797263696.34724 112 almost integer 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16235133991311881907338032643044452802932428234787601417304166592301609929567611017874830293973748714363590084283668417308148101106520292361975061079561047469069669896906073761876358816402423087300575017384369590147797240707648375531622320896078804446154187805466344721631282144290630300483195600524.84459 156 3653886178399625243812442730999484119324407061199933147807976079375653071438899177450573129322404111005507881331131643516239447979048497158368196323256390104855469886767561507306152001636194017056103393187182141228161643495982917194676297916242749065422653559965557272297357023851601055649202917502578.65806 157 827616624719328896744907209058776918488781262156240644152032589316310497917704278573041113320597973119052610891389955361675777280300864839798359318495709104405786183612422601907783121589048096310881843679773121995685487830798002687709349322579493477773619254140840812652652469623324919939156489093831587.22972 158 188651747237894417613400784714319430164487181309579572110462632930512663014657539519344465377549109559228618759073925619411673901972744772786479386562066146971411513364087599050195564367675620465382731491389020259045701812198099022750166591979801131774098298537336944673630511186276533671731001239967223800.23746 159 almost integer 43274543491028607741803469061010767430332337084204918320429970472522761733456850206230925917936911783068676793548484810501547707312243095701424289819741084975826041304289858217729559322645951397447088012570536933829919072607637917254177914793769476183640402383097529542299702743634513731188261671332932254628.07663 160 9989115086598518803349681088519177460149178912488412410905059070244738604774689634274070344436555109402659279232110783535274490750358665151829196799179716495498826260758985902415268744476672973019520046160095133088274392471418309497263207396112897853003214776015310211027892792035832782966576587524663300243062.25586 161 2320210734527074624586011355806062683946622201634170565761132603458136065426380617916539493562484900112525539171909627978684218622477066387351352845528355000246544687019145818030193274227941652190418350735454909748123225319064463065002910202358153633808321795173996711047052884364401107319960844624662750865561963.20776 162 542271756324167059007254730978145324569124764875811981803578051577656245145879252641769859044467735573174592878116902045688747745079890277328347439794926681405621617981500180835424211252093255259903866534501865120685941658015737501895542311796479627598797326235343292210998561711225288780556898235001591216862725682.35473 163 127520242107072944009636177539130894358409589467102276260639790163371648861969709950516325279197817888285773926484472321283834216795388524136788659362779131237642675745397364042721604539976866655177493575034472745318348010012382781078243724022664220550101966613289531158142222329055683803057796155184173435861289973823.16542 164 30171542627736794724169612701427008582854369260042238387694742648058009964996834533589653680196239838317041984854026336281448496304993967416452879847212774724439333084375639673369302753662088142090325545010483165187174455160308639000197727945458754444704770401110185737095370647953529247377682732926103776720308628000488.33801 165 7182175262625963189763104517731807198163649605294105861716403612516532133687002375109909822741597959894153507370918409234589663667969892331293771579535970230044905359683702097584790839563641516580452271152663054524797454555329441341659482619876707261768739390251005189046412826383903234673140562013460009392441624262637220.51842 166 1720040313274853666827115335939091650739208250799593023794829861952587626343860099432952148698888990897295690628134387270888413342083800245117174234831381625095854557797716804813993012909304429681208415451198686479714240349038342604241143810569834775743662808633598741043340892803617600895299198196064399721817770044370505112.32434 167 414409436225151983403603852149304547581842164354292978467632632560401997303344674386242278831025612457704277528075242210607785266248344344663661851587833561570013662143232324419788688671014374221536333447219660646579022588058146410304314573551354314922876516430033902532996487838990672869313297667368392570699069082842853436153.43428 168 100441561674655809580788161810936965607189711720727416661659064246699501831104282085382730092104867454351751579064156729879126294628818833387603865089866762699959202878446326748308584017662836802828135227244735619776439091590813877717579820222829492638893857207642330137085201207594855199600919206851477807987505851278970154472451.57729 169 24489205754691544623264832329594550104750853925618866853382162849683652515715090977664067926480897235302073129364480701709338396134050340013671850496093926231108425879351359460445875226843014928838964575168673581203100915797227282539217075185492794469982160707903045326677772156317343177126999667581083264403045609175732466087110246.24208 170 6006177468592502511355791298395715124982596151228973649401523395200900647133762244772091883935027971890070865975758343168759452961489649354858915118944128747601433827214419713707475109799941113127161984233038247249072766616399402313589555085097246982564733515792318111675980988024732784902601817651552719755556317693182733797695765534.51567 171 1481729098717000725283529050885774872123071341812319576351243601747285494557142539758067809356317663991602721366785252212373620468392461313042826938754215037253953762298790184170482669666995837879660248346430492047604605587870642265092964724183636165425471369178296552024714764261444814501846303803630123301434196854497920587387583186006.24709 172 almost integer 367681514297501124448715601012426383418150674500888613542161292058441474746316292257972416503541790335661531397785637711246410802747777691146646712451135574736900873957913842341015573640963581196399239954444636805487102152839329298006952033850848583741621812385868444034816812174342274343569782647775261745518433501670112266497462622652825.08196 173 91768247433514040734886437712739826078327056113306090031027099378379697640778177531200378293326294001012074268214026085595706915762156548630348782467866354607272025890590788082362564218263450123188287557021163420246318837169541073750480646617268384295217290699246972624245520821118907213537078219699714335106753896811467034738897315692936899.58903 174 23036485614111956746027162683618244602188820378803609032834216340093032087791810103306425439855234901089463246693708577281212859686726654730377311932172439126837982309765376053643189697657268158562871152504039902114886290015033818044134122625123278803570210856199759147225205941101842897498644098638248187067196282046465293313978644913571530147.59048 175 5816059122123121680778686393841149006273979824960504151711449238828259896339226553746785362471111776362242260455372409039557066074932750066305507622517682852364746763784297116912561779957570132184065907900619877187659024982961072877956548847161227025770393481155605345571132796031195822949077909287637273416921421363974482535299551785573674582078.72344 176 almost integer 1476780738928712034890198147146225599087596218552173442905221528436311623561444583662020128478087399864056039375838911024176231719307227165874467990904946775525650858998573905129478687474298711472839349006770880844086171038332612991615888598251568768584342847973282448668573705812314159657483183512324566105008304534750188963184100605497214008671621.07247 177 377106331990304149294896872050092930132116474753599922954529326802619413622971225766742702548754608609454708009827853436034438560140180540820383942469119841599328216836126996739352417299859793224817469350629506481034280275531770417163692882611783311386501693801643067930256050022975052203140081254231931736809926089274886361570213956235894636325499176.67892 178 96840799438942516030204230592888799221679367600797051179685806535473753548848641415136212271487572370153155280101458026640117710046285694419846676046547800720261388350881018230216261724286852209164827386783490240838364598265378754342229605496863591198587321905187604954540103259350234581106261635706160273139601897912303347023476384552515588948963571021.64996 179 25008401658025028919504962351581650606718047972297435681206563234864966573472931972036844140298996719126410658861248255504209669079596565082464939724765420806656242814938544134981993550186199251246672467872316851725281753767080796912780228328476467753179298309639610719108374758363655571985655559384063598204751405597578996842966497571138748727170348735310.64768 180 6494309469466567086342579845166376654935638986889609805298460864216507262095371598194373427571474926401222709972747664258580761067238049192677039572270457032465612687641160178983698173005216905899553958944123869822741100156768013260839027819619655090511856209219523061125380814521071117015048386970255272717052892881640243860526925509274745902515782132164029.76749 181 1695844759873174877285413377396817112669976741852260874200735707930334414192385993144414320988374494074378133115407091196613356125414993181046323854465020908495904895147058337509986843627730499298353430773472189950871794159444657661392179137667734766539478680306504586964527022381980737868623219614528428801921341705267857071997284981919738882437758101800013566.23523 182 445278802183954713629674174603551191654793853953148417270972255830999255160587007015932893701984750316171858905551248455842900867393174289396318458257456649020562122497170030803606592059446271396777115344453280860683671509737991107400289350554758286057586837050548246295079508314561202637063386195384236520417911229084708300685612840401045338651700916470083111481.39278 183 117559478109449762529952441563012971701433703710113343869704738204429351330990686746535747417829590052672540165173788500956291558260467455659340936798584747326874941226279758505310846091173621461084852500063050676770262026365269184990837020957169217121928771299228077617085981609324939863489293440558300157592667251794268189151172827613899261547989348309284302129465.23738 184 31206855598351599571613896444740010235823585807731722934156408627299605903464873284476090077121160984024354654267890077886380630249203258036129476015582192583529852444687280126694739379960323785325984725662202495624155397654135410622420806707802094572556751729398572449684179554529600423613022435251636670238111014121333403517490953093478147969463086642979756276284930.41255 185 8329065525493769949290655372038740278137351458379387272349672875965256974983771970375346747860499359955301342221761168768456057562657886823790331948991674160490509197405680200830606364480811167749317168640189980434755459343147250606084552601443594914422160458120218410355324926949800092659839971905367805947579718202980213757649571918539556571055577265322074490622441712.56721 186 2235032084369650726312204826316402353778555601746014369964473249023457712114934392384632268442469841076649647640300893152103005353054229432313330332672250455274590082608407771793758050779915654193457402038846489936505607515051335018178771316058131098014777471217386823496003058312198413572725358633469583468689482568351970425927432579084454047967667131694944066288632557211.25494 187 602975834713042034565983760390962608446431303895722457995414678890188031470369266019780074929626688611457349647474497917267528648892995102545678928041346048494399199264588012170290495187441392359823534205306765303476437562787104379493419043960492139460205528325742158264624472132202641785816589155924637114889169577288885528885039064855426102173623196865078272518812522781388.18523 188 almost integer 163543126345081135602466137557749797590586206478282853905652970085911912813158883884068533840548392490115699283608583721860533629613601753080576003315822331750679238035668377926389844065259324423929794560628523358971363428091752799784920258264095085234001939293781297844267730989922619757055777459014482005577310277003828763433825158703123653048113733432476739802104790585163697.07762 189 44593200039060364402102763699797035075668618780430757051572667376752268447817349238487814866546473731608722211688933951926851146905998770106131622809571161242441299360784008808002750154987508653648044543859518823284823573194254364938185486290451148539674806392621473401832449669703857953258707619516982055147506880075692250258927074876203616210272628015827010314526899890875372763.40413 190 12223533825207164236709289628526713711069819006290621785574070251194627588208132432145236203776132867004016048262192760092224738967672588039432329243044750419905108012710044731764354402919010639476252488807281881811659391434938893283071701692564866778899674179465512541450467047465701698223156593453222314161976352369107570568436392889020183382101259313707110811949466236803131876690.59385 191 3368252841667091525336254561443385846396301564470520256678533308760465357354817346323732580835898766758914068846133814314211303682039257324794096983662063419228045403527024400705335469880788169232561608744965635532537199883362293010613618563333514597666656553765101973336837059394780382142727422589059170040675086965790207256344779802469598528724647899509576935928934277023431740544057.78753 192 almost integer 932997440857588185178900869059757369538699967004525686039278814096413151816173030647842747305719921637293430570379820797507833232778321392034195309952284583489560078533696809574016963567760015515665419179267879307962345099511334826340353301910367705105827822710052806327089212933909151983339689407897810158756292306505647881853532693367832940591380018181127870675633979870981783490894069.99807 193 259783940749855926204189267140666406648171997016362190076968024884191920478871863745049203419286917117047865500302802662670740620368419829656778308248004754077183230330662278572677059682358974922306973068691887802341833208261510955958920871771985300506145447764082200812075032916301076421225725345863862138486689778400620147987819478193432082296181248984982596711555129842753347115931149703.67529 194 72709066586347431863579495206220499035405827729562949784089657445038184675939091263524059320871387677161439172120061124957114929788756760510401924340830740894355500172745495949404576950296521181414894394095194671955046816327059445092851798945679004021616461986863451490409473076458605570450859236721071700831215540774296947468474322510928650085713992079160373287984099781702113981409085350037.17803 195 20454916909398829892752215164920989897208967508911145569541639071798681151367986151586637417168878982759356217949945115064965970993199375332927173439923870829104067936753896348357732166411910443832892811336800002927524585573115237023068345456169550408501590655925574970453057389125931654011665876389108586726951383759931974258179854734476370812870633833368156050966001274942172376159862046835677.50930 196 5784000608649156804619652565433848911508149743675643946536150725436532389116172322612741989340830339373306970661331751696655753418948276412526377712932882811326796973392139574256671911388075367805652556711180740588202536114918285200690316037231737291427576180606631821287075764855299754138094663119369564089069749821374920588257646142446777715914950624540907820274843935902766556895110704576908045.69749 197 1643876151935586248754399939921773116810182486644547522574931394691560184496243423318050284392287070630741841933378019077143506399048999971329130724719290951693185394006413229467436103698095895857668552124021504010516518577784964721473716632131171833401903867648995563495500287568657134216864406410565598048745050529812285114555953856738899125594156828764334624325053726883824339337487192462825547564.49176
version 3
This REXX version takes advantage that the Hickerson series are computed in order, so that the factorials need not be (re-)computed from scratch. <lang rexx>/*REXX program to calculate and show the Hickerson series (are near integer). */ numeric digits length(ln2())-length(.) /*be able to calculate big factorials. */ parse arg N . /*get optional number of values to use.*/ if N== then N= 1 /*Not specified? Then use the default.*/ != 1 /* [↓] compute possible Hickerson #s. */
do j=1 for N; #= Hickerson(j) /*traipse thru a range of Hickerson #s.*/
?= right(# * 10 % 1, 1) /*get 1st decimal digit past dec. point*/
if ?==0 | ?==9 then _= 'almost integer' /*the number is, */
else _= ' ' /* or it ain't. */
say right(j,3) _ format(#, , 5) /*show number with 5 dec digs fraction.*/
end /*j*/
exit /*stick a fork in it, we're all done. */ /*─────────────────────────────────────────────────────────────────────────────────────────────────────────*/ Hickerson: parse arg z; != ! * z; return ! / (2*ln2() ** (z+1) ) ln2: return .69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641,
|| 8687542001481020570685733685520235758130557032670751635075961930727570828371435190307038623891673471123, || 3501153644979552391204751726815749320651555247341395258829504530070953263666426541042391578149520437404, || 3038550080194417064167151864471283996817178454695702627163106454615025720740248163777338963855069526066, || 83411372738737229289564935470257626520988596932019650585547647033067936544325476327449512504060694381470</lang>
- output is identical to the 2nd REXX version.
Ring
<lang ring> n = 12 hick = 0 decimals(8)
for i = 1 to n
see "h(" + string(i) + ") = "
see "" + hickersonSeries(i) + " "
if nearly(hick) = 1 see "nearly integer" + nl
else see "not nearly integer" + nl ok
next
func hickersonSeries nr
hick = fact(nr) / (2 * pow(log(2), nr+1))
return hick
func fact nr if nr = 1 return 1 else return nr * fact(nr-1) ok
func nearly nr
strNr = string(nr)
sub = substr(strNr,".")
sub2 = substr(strNr,sub+1,1)
if (sub2 = "0" or sub2="9") return 1
else return 0 ok
return sub
</lang>
Ruby
Using the BigDecimal standard library: <lang ruby>require "bigdecimal"
LN2 = BigMath::log(2,16) #Use LN2 = Math::log(2) to see the difference with floats FACTORIALS = Hash.new{|h,k| h[k] = k * h[k-1]} FACTORIALS[0] = 1
def hickerson(n)
FACTORIALS[n] / (2 * LN2 ** (n+1))
end
def nearly_int?(n)
int = n.round n.between?(int - 0.1, int + 0.1)
end
1.upto(17) do |n|
h = hickerson(n)
str = nearly_int?(h) ? "nearly integer" : "NOT nearly integer"
puts "n:%3i h: %s\t%s" % [n, h.to_s('F')[0,25], str] #increase the 25 to print more digits, there are 856 of them
end</lang>
- Output:
n: 1 h: 1.04068449050280389893479 nearly integer n: 2 h: 3.00278070715690544349976 nearly integer n: 3 h: 12.9962905052769664622248 nearly integer n: 4 h: 74.9987354476616001276345 nearly integer n: 5 h: 541.001518516423507569202 nearly integer n: 6 h: 4683.00124726225743718046 nearly integer n: 7 h: 47292.9987313146239048228 nearly integer n: 8 h: 545834.997907485167067291 nearly integer n: 9 h: 7087261.00162289912097918 nearly integer n: 10 h: 102247563.005271042011088 nearly integer n: 11 h: 1622632572.99755004985287 nearly integer n: 12 h: 28091567594.9815724407151 nearly integer n: 13 h: 526858348381.001248286180 nearly integer n: 14 h: 10641342970443.0845319270 nearly integer n: 15 h: 230283190977853.037436039 nearly integer n: 16 h: 5315654681981354.51307674 NOT nearly integer n: 17 h: 130370767029135900.457985 NOT nearly integer
Scala
Functional Programming ♫
<lang Scala>import scala.annotation.tailrec
object Hickerson extends App {
def almostInteger(n: Int): Boolean = {
def ln2 = BigDecimal("0.69314718055994530941723212145818")
def div: BigDecimal = ln2.pow(n + 1) * 2
def factorial(num: Int): Long = {
@tailrec
def accumulated(acc: Long, n: Long): Long =
if (n <= 0) acc else accumulated(acc * n, n - 1)
accumulated(1, num) }
((BigDecimal(factorial(n)) / div * 10).toBigInt() % 10).toString.matches("0|9")
}
val aa = (1 to 17).map(n => n -> almostInteger(n))
println(s"Function h(n) gives a almost integer with a n of ${aa.filter(_._2).map(_._1).mkString(", ")}.")
println(s"While h(n) gives NOT an almost integer with a n of ${aa.filter(!_._2).map(_._1).mkString(", ")}.")
}</lang>
- Output:
See it in running in your browser by ScalaFiddle (JavaScript executed in browser)
or by Scastie (remote JVM).
Seed7
<lang seed7>$ include "seed7_05.s7i";
include "bigrat.s7i";
const proc: main is func
local
const bigRational: ln2 is 6931471805599453094172_ / 10000000000000000000000_;
var bigRational: h is 1_ / 2_ / ln2;
var integer: n is 0;
var string: stri is "";
begin
for n range 1 to 17 do
h := h * bigRational conv n / ln2;
stri := h digits 3;
writeln(n lpad 2 <& stri lpad 23 <& " is almost integer: " <&
stri[succ(pos(stri, '.'))] in {'0', '9'});
end for;
end func;</lang>
- Output:
1 1.041 is almost integer: TRUE 2 3.003 is almost integer: TRUE 3 12.996 is almost integer: TRUE 4 74.999 is almost integer: TRUE 5 541.002 is almost integer: TRUE 6 4683.001 is almost integer: TRUE 7 47292.999 is almost integer: TRUE 8 545834.998 is almost integer: TRUE 9 7087261.002 is almost integer: TRUE 10 102247563.005 is almost integer: TRUE 11 1622632572.998 is almost integer: TRUE 12 28091567594.982 is almost integer: TRUE 13 526858348381.001 is almost integer: TRUE 14 10641342970443.085 is almost integer: TRUE 15 230283190977853.037 is almost integer: TRUE 16 5315654681981354.513 is almost integer: FALSE 17 130370767029135900.458 is almost integer: FALSE
Sidef
<lang ruby>func h(n) {
n! / (2 * pow(2.log, n+1))
} { |n|
"h(%2d) = %22s is%s almost an integer.\n".printf(
n, var hn = h(n).round(-3), hn.to_s ~~ /\.[09]/ ? : ' NOT')
} << 1..17</lang>
- Output:
h( 1) = 1.041 is almost an integer. h( 2) = 3.003 is almost an integer. h( 3) = 12.996 is almost an integer. h( 4) = 74.999 is almost an integer. h( 5) = 541.002 is almost an integer. h( 6) = 4683.001 is almost an integer. h( 7) = 47292.999 is almost an integer. h( 8) = 545834.998 is almost an integer. h( 9) = 7087261.002 is almost an integer. h(10) = 102247563.005 is almost an integer. h(11) = 1622632572.998 is almost an integer. h(12) = 28091567594.982 is almost an integer. h(13) = 526858348381.001 is almost an integer. h(14) = 10641342970443.085 is almost an integer. h(15) = 230283190977853.037 is almost an integer. h(16) = 5315654681981354.513 is NOT almost an integer. h(17) = 130370767029135900.458 is NOT almost an integer.
Tcl
<lang tcl>package require math::bigfloat namespace import math::bigfloat::*
- Precision is an arbitrary value large enough to provide clear demonstration
proc hickerson {n {precision 28}} {
set fac 1
for {set i 1} {$i <= $n} {incr i} {set fac [expr {$fac * $i}]}
set numerator [int2float $fac $precision]
set 2 [int2float 2 $precision]
set denominator [mul $2 [pow [log $2] [expr {$n + 1}]]]
return [tostr -nosci [div $numerator $denominator]]
}
for {set n 1} {$n <= 17} {incr n} {
set h [hickerson $n]
set almost [regexp {\.[09]} $h]
puts [format "h(%d) = %s (%salmost integer)" $n $h [expr {$almost ? "" : "not "}]]
}</lang>
- Output:
h(1) = 1.040684490502803898934790802 (almost integer) h(2) = 3.00278070715690544349976741 (almost integer) h(3) = 12.9962905052769664622248845 (almost integer) h(4) = 74.998735447661600127634550 (almost integer) h(5) = 541.00151851642350756920277 (almost integer) h(6) = 4683.001247262257437180467 (almost integer) h(7) = 47292.9987313146239048228355 (almost integer) h(8) = 545834.99790748516706729104 (almost integer) h(9) = 7087261.0016228991209791875 (almost integer) h(10) = 102247563.005271042011088389 (almost integer) h(11) = 1622632572.99755004985287486 (almost integer) h(12) = 28091567594.981572440715189 (almost integer) h(13) = 526858348381.00124828618049 (almost integer) h(14) = 10641342970443.0845319270951 (almost integer) h(15) = 230283190977853.037436039126 (almost integer) h(16) = 5315654681981354.5130767435 (not almost integer) h(17) = 130370767029135900.457985349 (not almost integer)
TI-83 BASIC
<lang ti83b>For(N,1,17 N!/(2ln(2)^(N+1→H Disp N,H,"IS round(H,1)-iPart(H) If not(Ans=.9 or not(Ans Disp "NOT Disp "ALMOST INTEGER End</lang> (untested)
Wren
This is a tricky task for Wren which doesn't have arbitrary precision float or decimal but does have (via the above module) BigInt.
I've therefore used the most accurate value I could find for log2 (63 digit accuracy), represented this as a BigInt, and worked from there. <lang ecmascript>import "/math" for Int import "/fmt" for Fmt import "/big" for BigInt
var hickerson = Fn.new { |n|
var fact = BigInt.new(Int.factorial(n)) // accurate up to n == 18
var ln2 = BigInt.new("693147180559945309417232121458176568075500134360255254120680009") // 63 digits
var mult = BigInt.new("1e64").pow(n+1) // 64 == ln2 digit count + 1
return fact * mult /(BigInt.two * ln2.pow(n+1))
}
System.print("Values of h(n), truncated to 1 dp, and whether 'almost integers' or not:") for (i in 1..17) {
var h = hickerson.call(i).toString
var hl = h.count
var k = hl - i - 1
var ai = (h[k] == "0" || h[k] == "9")
var s = h[0...k] + "." + h[k]
Fmt.print("$2d: $20s $s", i, s, ai)
}</lang>
- Output:
Values of h(n), truncated to 1 dp, and whether 'almost integers' or not: 1: 1.0 true 2: 3.0 true 3: 12.9 true 4: 74.9 true 5: 541.0 true 6: 4683.0 true 7: 47292.9 true 8: 545834.9 true 9: 7087261.0 true 10: 102247563.0 true 11: 1622632572.9 true 12: 28091567594.9 true 13: 526858348381.0 true 14: 10641342970443.0 true 15: 230283190977853.0 true 16: 5315654681981354.5 false 17: 130370767029135900.4 false
Since the above solution was posted, support for arbitrary precision rational numbers has been added to the Wren-big module. The BigRat class has methods to convert to and from decimal number representation. However, it doesn't support transcendental functions and so again I've used the most accurate value I could find for log2 and represented it as a BigRat.
The following produces the same results as before though is slower than the BigInt version due to the implicit conversions needed. <lang ecmascript>import "/math" for Int import "/fmt" for Fmt import "/big" for BigRat
var hickerson = Fn.new { |n|
var fact = BigRat.new(Int.factorial(n))
var ln2 = BigRat.fromDecimal("0.693147180559945309417232121458176568075500134360255254120680009")
return fact / (BigRat.two * ln2.pow(n+1))
}
System.print("Values of h(n), truncated to 1 dp, and whether 'almost integers' or not:") for (i in 1..17) {
var h = hickerson.call(i).toDecimal(1, false)
var hl = h[-1]
var ai = (hl == "0" || hl == "9")
Fmt.print("$2d: $20s $s", i, h, ai)
}</lang>
zkl
Uses lib GMP (integer) and some fiddling to fake up the floating point math. <lang zkl>var [const] BN=Import("zklBigNum"),
X =BN("1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"),
ln2X=BN("693147180559945309417232121458176568075500134360255254120680009493393621969694715605863326996418687")
;
fcn hickerson(n){ BN(n).factorial()*X.pow(n+1)*10/2/ln2X.pow(n+1) }</lang> <lang zkl>foreach n in ([1..18]){
hs,ld,isH:=hickerson(n).toString(),hs[-1],"90".holds(ld);
println("h(%2d)=%s.%s almost integer: %b".fmt(n,hs[0,-1],ld,isH));
}</lang>
- Output:
h( 1)=1.0 almost integer: True h( 2)=3.0 almost integer: True h( 3)=12.9 almost integer: True h( 4)=74.9 almost integer: True h( 5)=541.0 almost integer: True h( 6)=4683.0 almost integer: True h( 7)=47292.9 almost integer: True h( 8)=545834.9 almost integer: True h( 9)=7087261.0 almost integer: True h(10)=102247563.0 almost integer: True h(11)=1622632572.9 almost integer: True h(12)=28091567594.9 almost integer: True h(13)=526858348381.0 almost integer: True h(14)=10641342970443.0 almost integer: True h(15)=230283190977853.0 almost integer: True h(16)=5315654681981354.5 almost integer: False h(17)=130370767029135900.4 almost integer: False h(18)=3385534663256845326.3 almost integer: False
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