Minimum multiple of m where digital sum equals m
Generate the sequence a(n) when each element is the minimum integer multiple m such that the digit sum of n times m is equal to n.
- Task
- Find the first 40 elements of the sequence.
- Stretch
- Find the next 30 elements of the sequence.
- See also
ALGOL 68
<lang algol68>BEGIN # find the smallest m where mn = digit sum of n, n in 1 .. 70 #
# returns the digit sum of n, n must be >= 0 #
OP DIGITSUM = ( INT n )INT:
IF n < 10 THEN n
ELSE
INT result := n MOD 10;
INT v := n OVER 10;
WHILE v > 0 DO
result +:= v MOD 10;
v := v OVER 10
OD;
result
FI # DIGITSUM # ;
# show the minimum multiples of n where the digit sum of the multiple is n #
FOR n TO 70 DO
BOOL found multiple := FALSE;
FOR m WHILE NOT found multiple DO
IF DIGITSUM ( m * n ) = n THEN
found multiple := TRUE;
print( ( " ", whole( m, -8 ) ) );
IF n MOD 10 = 0 THEN print( ( newline ) ) FI
FI
OD
OD
END</lang>
- Output:
1 1 1 1 1 1 1 1 1 19
19 4 19 19 13 28 28 11 46 199
19 109 73 37 199 73 37 271 172 1333
289 559 1303 847 1657 833 1027 1576 1282 17497
4339 2119 2323 10909 11111 12826 14617 14581 16102 199999
17449 38269 56413 37037 1108909 142498 103507 154981 150661 1333333
163918 322579 315873 937342 1076923 1030303 880597 1469116 1157971 12842857
Haskell
<lang haskell>import Data.Bifunctor (first) import Data.List (elemIndex, intercalate, transpose) import Data.List.Split (chunksOf) import Data.Maybe (fromJust) import Text.Printf (printf)
A131382 ------------------------
a131382 :: [Int] a131382 =
fromJust . (elemIndex <*> productDigitSums) <$> [1 ..]
productDigitSums :: Int -> [Int] productDigitSums n = digitSum . (n *) <$> [0 ..]
TEST -------------------------
main :: IO () main =
(putStrLn . table " ") $ chunksOf 10 $ show <$> take 40 a131382
GENERIC ------------------------
digitSum :: Int -> Int digitSum 0 = 0 digitSum n = uncurry (+) (first digitSum $ quotRem n 10)
table :: String -> String -> String table gap rows =
let ws = maximum . fmap length <$> transpose rows
pw = printf . flip intercalate ["%", "s"] . show
in unlines $ intercalate gap . zipWith pw ws <$> rows</lang>
- Output:
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333 289 559 1303 847 1657 833 1027 1576 1282 17497
J
Implementation:
<lang J> findfirst=: Template:($:@((+1+i.@+:)@
A131382=: {{y&Template:X = sumdigits x*y findfirst}}"0
sumdigits=: +/@|:@(10&#.inv)</lang>
Task example: <lang J> A131382 1+i.4 10
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333
289 559 1303 847 1657 833 1027 1576 1282 17497</lang>
Julia
<lang julia>minproddigsum(n) = findfirst(i -> sum(digits(n * i)) == n, 1:typemax(Int32))
for j in 1:70
print(lpad(minproddigsum(j), 10), j % 7 == 0 ? "\n" : "")
end
</lang>
- Output:
1 1 1 1 1 1 1
1 1 19 19 4 19 19
13 28 28 11 46 199 19
109 73 37 199 73 37 271
172 1333 289 559 1303 847 1657
833 1027 1576 1282 17497 4339 2119
2323 10909 11111 12826 14617 14581 16102
199999 17449 38269 56413 37037 1108909 142498
103507 154981 150661 1333333 163918 322579 315873
937342 1076923 1030303 880597 1469116 1157971 12842857
Perl
<lang perl>#!/usr/bin/perl
use strict; # https://rosettacode.org/wiki/Minimum_multiple_of_m_where_digital_sum_equals_m use warnings; use ntheory qw( sumdigits );
my @answers = map
{
my $m = 1;
$m++ until sumdigits($m*$_) == $_;
$m;
} 1 .. 70;
print "@answers\n\n" =~ s/.{65}\K /\n/gr;</lang>
- Output:
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333 289 559 1303 847 1657 833 1027 1576 1282 17497 4339 2119 2323 10909 11111 12826 14617 14581 16102 199999 17449 38269 56413 37037 1108909 142498 103507 154981 150661 1333333 163918 322579 315873 937342 1076923 1030303 880597 1469116 1157971 12842857
Phix
with javascript_semantics integer c = 0, n = 1 while c<70 do integer m = 1 while true do integer nm = n*m, t = 0 while nm do t += remainder(nm,10) nm = floor(nm/10) end while if t=n then exit end if m += 1 end while c += 1 printf(1,"%-8d%s",{m,iff(remainder(c,10)=0?"\n":"")}) n += 1 end while
- Output:
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333 289 559 1303 847 1657 833 1027 1576 1282 17497 4339 2119 2323 10909 11111 12826 14617 14581 16102 199999 17449 38269 56413 37037 1108909 142498 103507 154981 150661 1333333 163918 322579 315873 937342 1076923 1030303 880597 1469116 1157971 12842857
Python
<lang python>A131382
from itertools import count, islice
- a131382 :: [Int]
def a131382():
An infinite series of the terms of A131382
return (
elemIndex(x)(
productDigitSums(x)
) for x in count(1)
)
- productDigitSums :: Int -> [Int]
def productDigitSums(n):
The sum of the decimal digits of n return (digitSum(n * x) for x in count(0))
- ------------------------- TEST -------------------------
- main :: IO ()
def main():
First 40 terms of A131382
print(
table(10)([
str(x) for x in islice(
a131382(),
40
)
])
)
- ----------------------- GENERIC ------------------------
- chunksOf :: Int -> [a] -> a
def chunksOf(n):
A series of lists of length n, subdividing the
contents of xs. Where the length of xs is not evenly
divisible, the final list will be shorter than n.
def go(xs):
return (
xs[i:n + i] for i in range(0, len(xs), n)
) if 0 < n else None
return go
- digitSum :: Int -> Int
def digitSum(n):
The sum of the digital digits of n. return sum(int(x) for x in list(str(n)))
- elemIndex :: a -> [a] -> (None | Int)
def elemIndex(x):
Just the first index of x in xs,
or None if no elements match.
def go(xs):
try:
return next(
i for i, v in enumerate(xs) if x == v
)
except StopIteration:
return None
return go
- table :: Int -> [String] -> String
def table(n):
A list of strings formatted as
right-justified rows of n columns.
def go(xs):
w = len(xs[-1])
return '\n'.join(
' '.join(row) for row in chunksOf(n)([
s.rjust(w, ' ') for s in xs
])
)
return go
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333 289 559 1303 847 1657 833 1027 1576 1282 17497
Raku
<lang perl6>sub min-mult-dsum ($n) { (1..∞).first: (* × $n).comb.sum == $n }
say .fmt("%2d: ") ~ .&min-mult-dsum for flat 1..40, 41..70;</lang>
- Output:
1: 1 2: 1 3: 1 4: 1 5: 1 6: 1 7: 1 8: 1 9: 1 10: 19 11: 19 12: 4 13: 19 14: 19 15: 13 16: 28 17: 28 18: 11 19: 46 20: 199 21: 19 22: 109 23: 73 24: 37 25: 199 26: 73 27: 37 28: 271 29: 172 30: 1333 31: 289 32: 559 33: 1303 34: 847 35: 1657 36: 833 37: 1027 38: 1576 39: 1282 40: 17497 41: 4339 42: 2119 43: 2323 44: 10909 45: 11111 46: 12826 47: 14617 48: 14581 49: 16102 50: 199999 51: 17449 52: 38269 53: 56413 54: 37037 55: 1108909 56: 142498 57: 103507 58: 154981 59: 150661 60: 1333333 61: 163918 62: 322579 63: 315873 64: 937342 65: 1076923 66: 1030303 67: 880597 68: 1469116 69: 1157971 70: 12842857
Wren
<lang ecmascript>import "./math" for Int import "./seq" for Lst import "./fmt" for Fmt
var res = [] for (n in 1..70) {
var m = 1 while (Int.digitSum(m * n) != n) m = m + 1 res.add(m)
} for (chunk in Lst.chunks(res, 10)) Fmt.print("$,10d", chunk)</lang>
- Output:
1 1 1 1 1 1 1 1 1 19
19 4 19 19 13 28 28 11 46 199
19 109 73 37 199 73 37 271 172 1,333
289 559 1,303 847 1,657 833 1,027 1,576 1,282 17,497
4,339 2,119 2,323 10,909 11,111 12,826 14,617 14,581 16,102 199,999
17,449 38,269 56,413 37,037 1,108,909 142,498 103,507 154,981 150,661 1,333,333
163,918 322,579 315,873 937,342 1,076,923 1,030,303 880,597 1,469,116 1,157,971 12,842,857
XPL0
<lang XPL0>func SumDigits(N); \Return sum of digits in N int N, S; [S:= 0; while N do
[N:= N/10; S:= S + rem(0); ];
return S; ];
int C, N, M; [C:= 0; N:= 1; repeat M:= 1;
while SumDigits(N*M) # N do M:= M+1;
IntOut(0, M);
C:= C+1;
if rem (C/10) then ChOut(0, 9\tab\) else CrLf(0);
N:= N+1;
until C >= 40+30; ]</lang>
- Output:
1 1 1 1 1 1 1 1 1 19 19 4 19 19 13 28 28 11 46 199 19 109 73 37 199 73 37 271 172 1333 289 559 1303 847 1657 833 1027 1576 1282 17497 4339 2119 2323 10909 11111 12826 14617 14581 16102 199999 17449 38269 56413 37037 1108909 142498 103507 154981 150661 1333333 163918 322579 315873 937342 1076923 1030303 880597 1469116 1157971 12842857