A003634 -id:A003634 - cp's OEIS frontend

A007471 Sum of digits of n a(n) is n ( = A003634/n), or 0 if no such number exists.

1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 18, 9, 9, 9, 9, 9, 9, 9, 6, 9, 18, 6, 9, 9, 6, 9, 9, 4, 9, 9, 12, 18, 18, 3, 9, 9, 3, 9, 9, 3, 18, 18, 12, 18, 9, 5, 9, 9, 9, 9, 18, 6, 18, 18, 2, 9, 9, 9, 9, 9, 12, 0, 0, 5, 0, 18, 3, 9, 9, 3, 18, 18, 7, 27, 0, 12, 18

Offset: 1

N. J. A. Sloane

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zeros inserted for consistency with A003634 by Sean A. Irvine, Jan 04 2018

A057147 a(n) = n times sum of digits of n.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 10, 22, 36, 52, 70, 90, 112, 136, 162, 190, 40, 63, 88, 115, 144, 175, 208, 243, 280, 319, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 160, 205, 252, 301, 352, 405, 460, 517, 576, 637, 250, 306, 364, 424, 486, 550, 616

Offset: 0

N. J. A. Sloane, Sep 13 2000

A056992(n) = A010888(a(n)). - Reinhard Zumkeller, Mar 19 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
F. B. Diniz, About a new family of sequences, arXiv:1607.06082 [math.GM], 2016.

Cf. A007953, A003634, A005349, A052489, A052490, A003635, A245788.

Iterations: A047892 (start=2), A047912 (start=3), A047897 (start=5), A047898 (start=6), A047899 (start=7), A047900 (start=8), A047901 (start=9), A047902 (start=11).

a057147 n = a007953 n * n  -- Reinhard Zumkeller, Mar 19 2014

for n from 0 to 150 do printf(`%d,`,n*add(convert(n, base, 10)[i], i=1..nops(convert(n,base, 10)))) od:

Table[n*Total[IntegerDigits[n]], {n, 0, 100}]

a(n) = n*sumdigits(n) \\ Franklin T. Adams-Watters, Aug 03 2014

[n*sum([int(d) for d in str(n)]) for n in range(10**5)] # Chai Wah Wu, Aug 05 2014

a(n) = n*A007953(n). - Michel Marcus, Aug 10 2014

G.f.: x * (d/dx) (1/(1 - x))*Sum_{k>=1} (x^k - x^(10^k+k) - 9*x^(10^k))/(1 - x^(10^k)). - Ilya Gutkovskiy, Mar 27 2018

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Sep 13 2000

A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

Original entry on oeis.org

62, 63, 65, 75, 84, 95, 161, 173, 195, 216, 261, 266, 272, 276, 326, 371, 372, 377, 381, 383, 386, 387, 395, 411, 416, 422, 426, 431, 432, 438, 441, 443, 461, 466, 471, 476, 482, 483, 486, 488, 491, 492, 493, 494, 497, 498, 516, 521, 522, 527, 531, 533, 536

Offset: 1

N. J. A. Sloane and Mira Bernstein

J. H. Conway, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Mondot, Table of n, a(n) for n = 1..10867
Giovanni Resta, Numbers Aplenty: Inconsummate numbers

Cf. A003634, A052489, A052490, A052491, A058898-A058917.

Maple
```
For Maple code see A058906.
```

nmax = 1000; Reap[ Do[k = n; kmax = 100*n; While[ Tr[ IntegerDigits[k]]*n != k && k < kmax, k = k + n]; If[k == kmax, Sow[n]], {n, 1, nmax}]][[2, 1]] (* Jean-François Alcover, Jul 12 2012 *)

from itertools import count, islice, combinations_with_replacement
def A003635_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 9*l*n < 10**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(10),l):
                if (s:=sum(d))>0 and sorted(str(s*n)) == [str(e) for e in d]:
                    break
            else:
                continue
            break
A003635_list = list(islice(A003635_gen(),20)) # Chai Wah Wu, May 09 2023

A052489 Largest number that is n times sum of its decimal digits.

Original entry on oeis.org

0, 9, 18, 27, 48, 45, 54, 84, 72, 81, 90, 198, 108, 195, 126, 135, 288, 153, 162, 399, 180, 378, 396, 207, 216, 375, 468, 486, 588, 261, 270, 558, 576, 594, 408, 315, 648, 999, 684, 351, 480, 738, 756, 774, 792, 405, 966, 846, 864, 882, 450, 918, 936, 954

Offset: 0

Henry Bottomley, Mar 16 2000

It is infinite, as pointed out by Dr. Geoffrey Landis: Clearly if you have one integer that is N times the sum of its decimal digits, then when you add a 0 to the end, you have an integer that is 10N times the sum of its decimal digits. - Jonathan Vos Post, Feb 06 2011

There are, however, positive n for which no positive number is n times the sum of its digits, in which case a(n) = 0, as for n = 62, 63, 65, 75, 84, ..., cf. A003635. - M. F. Hasler, Aug 24 2025

Daniel Mondot, Table of n, a(n) for n = 0..9999

Cf. A003634, A003635, A052490.

p[n_] := 10(Length[IntegerDigits[n]]+1); a[0]=0; a[n_] := Catch[For[k = p[n]*n, k >= 0, k--, If[k == n*Total[IntegerDigits[k]], If[k == 0, Print["a(", n, ") not found"]]; Throw[k]]]]; Table[a[n], {n, 0, 1000}]  (* Jean-François Alcover, Jul 19 2012 updated Oct 06 2016 after Daniel Mondot's observations *)

a(n) = {nbd = 1; while (9*nbd*n > 10^nbd, nbd++); forstep(k=9*nbd*n, 1, -1, if (sumdigits(k)*n == k, return(k));); 0;} \\ Michel Marcus, Oct 05 2016

A072081 Numbers divisible by the square of the sum of their digits in base 10.

Original entry on oeis.org

1, 10, 20, 50, 81, 100, 112, 162, 200, 243, 324, 392, 400, 405, 500, 512, 605, 648, 810, 972, 1000, 1053, 1100, 1120, 1134, 1183, 1215, 1296, 1400, 1620, 1701, 1900, 1944, 2000, 2025, 2106, 2156, 2240, 2268, 2300, 2401, 2430, 2511, 2592, 2704, 2800, 2916

Offset: 1

Labos Elemer, Jun 14 2002

If k is a term, then 10 * k is a term. There are an infinite number of terms that are not divisible by 10. The numbers m = 24 * 10^(42 * k - 40) +1, k >= 1, are divisible by 7^2 = digsum(m)^2. Also, the numbers s = 491 * 10^(42 * k - 8) + 3, k >= 1, are divisible by 17^2 = digsum(s)^2. - Marius A. Burtea, Mar 19 2020

The numbers 2^A095412(n), n >= 5, are terms. - Marius A. Burtea, Apr 02 2020

			k=9477, sumdigits(9477)=27, q=9477=27*27*13.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A003132, A005349, A003634.

[k:k in [1..3000]| k mod &+Intseq(k)^2 eq 0]; // Marius A. Burtea, Mar 19 2020

sud[x_] := Apply[Plus, IntegerDigits[x]] Do[s=sud[n]^2; If[IntegerQ[n/s], Print[n]], {n, 1, 10000}]
Select[Range[3000],Divisible[#,Total[IntegerDigits[#]]^2]&] (* Harvey P. Dale, May 04 2011 *)

for(n=1,10^4,s=sumdigits(n);if(!(n%s^2),print1(n,", "))) \\ Derek Orr, Apr 29 2015

def ok(n): return n and n%sum(di for di in map(int, str(n)))**2 == 0
print([k for k in range(3000) if ok(k)]) # Michael S. Branicky, Jan 10 2025

A056770 Smallest number that is n times the product of its digits or 0 if impossible.

Original entry on oeis.org

1, 36, 15, 384, 175, 12, 735, 128, 135, 0, 11, 1296, 624, 224, 0, 0, 816, 216, 1197, 0, 315, 132, 115, 0, 0, 0, 2916, 1176, 3915, 0, 93744, 0, 51975, 78962688, 0, 82944, 1184, 0, 0, 0, 31488, 0, 0, 77616, 77175, 4416, 0, 12288, 1715, 0, 612

Offset: 1

Robert G. Wilson v, Aug 16 2000

			a(4) = 384 because 4*(product of digits of 384) = 4*96 = 384, and no number smaller than 384 has this property.

David W. Wilson, Table of n, a(n) for n = 1..1000

Cf. A007954, A007602, A003634.

Do[k = n; If[Mod[n, 10] == 0, Print[0]; Continue[]]; While[Apply[Times, RealDigits[k][[1]]]*n != k, k += n]; Print[k], {n, 1, 14}]

from itertools import count, combinations_with_replacement
from math import prod
def A056770(n):
    if not n%10: return 0
    for l in count(1):
        if 9**l*n < 10**(l-1): return 0
        c = 10**l
        for d in combinations_with_replacement(range(1,10),l):
            if sorted(str(a:=prod(d)*n)) == list(str(e) for e in d):
                c = min(c,a)
        if c < 10**l:
            return c # Chai Wah Wu, May 09 2023

a(15) onwards from David W. Wilson, Jan 20 2016

A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 6, 4, 10, 12, 21, 8, 18, 20, 55, 24, 0, 42, 60, 16, 34, 36, 0, 40, 126, 110, 69, 48, 0, 0, 81, 84, 116, 120, 155, 32, 66, 68, 0, 72, 185, 0, 156, 80, 205, 252, 172, 220, 180, 138, 0, 96, 0, 0, 204, 0, 212, 162, 0, 168, 228, 232, 295, 240, 366, 310, 378, 64, 130

Offset: 1

Henry Bottomley, Nov 26 2001

a(n) is bounded above by n*A272756(n), so a program only has to check values up to that point to see if a(n) is zero. - Peter Kagey, May 05 2016

			a(23) is 69 since 69 is written in binary as 1000101, 69/(1+0+0+0+1+0+1)=23 and there is no smaller possibility (neither 23 nor 46 are divisible by their number of binary 1's).

Peter Kagey, Table of n, a(n) for n = 1..10000

A003634 is the base-10 equivalent.

Cf. A000120, A049445, A058898, A065413, A065878, A065880, A272756.

Table[SelectFirst[Range[2^12], # == n First@ DigitCount[#, 2] &] /. k_ /; MissingQ@ k -> 0, {n, 80}] (* Michael De Vlieger, May 05 2016, Version 10.2 *)

A037478 Number of positive solutions to "numbers that are n times sum of their digits".

Original entry on oeis.org

9, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 11, 1, 1, 3, 1, 1, 3, 2, 2, 12, 1, 1, 3, 1, 1, 4, 1, 2, 15, 2, 1, 4, 1, 1, 3, 1, 1, 13, 2, 2, 3, 1, 1, 4, 1, 1, 13, 1, 1, 2, 1, 1, 3, 0, 0, 7, 0, 1, 4, 1, 1, 4, 1, 1, 8, 1, 0, 3, 1, 1, 4, 1, 1, 10, 1, 0, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 0, 1, 3, 1, 1, 9, 1

Offset: 1

Henry Bottomley, Sep 12 2000

It appears that the largest terms occur when n=1 mod 9 and moderately large terms when n=4 or 7 mod 9.

			a(13)=3 since the only three solutions are 117=9*13, 156=12*13 and 195=15*13.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A003634, A003635, A005349, A052489, A052490, A057147, A058913.

read("transforms"):
A037478 := proc(n)
    local a,x,k;
    a := 0 ;
    for k from 1 do
        x := n*k ;
        if digsum(x)*n = x then
            a := a+1 ;
        end if;
        # may stop if x/digsum(x)>n, so if x/#digits(x) > 9*n
        if x/A055642(x) > 9*n then
            break;
        end if;
    end do:
    a ;
end proc:
seq(A037478(n),n=1..101) ; # R. J. Mathar, May 11 2016

A052490 Numbers n with only one nonzero solution to "numbers that are n times sum of their digits".

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 29, 30, 32, 33, 35, 39, 41, 42, 44, 45, 50, 51, 53, 54, 56, 57, 59, 60, 66, 68, 69, 71, 72, 74, 77, 78, 80, 81, 83, 86, 87, 89, 90, 92, 93, 96, 98, 99, 101, 102, 104, 105, 108, 110, 111, 113, 114, 117, 119, 120, 122

Offset: 1

Henry Bottomley, Mar 16 2000

			a(2)=3 since there is only one positive number which is three times the sum of its digits, namely 27=3*9

Cf. A003634, A052489, A003635.

A072083 Numbers divisible by the 4th power of the sum of their digits in base 10.

Original entry on oeis.org

1, 10, 100, 1000, 2000, 2401, 5000, 10000, 13122, 20000, 24010, 50000, 100000, 110000, 131220, 140000, 190000, 200000, 230000, 234256, 240100, 280000, 320000, 370000, 390625, 400221, 410000, 460000, 500000, 512000, 550000, 614656, 640000

Offset: 1

Labos Elemer, Jun 14 2002

If k is a term, then 10*k is a term. There are an infinite number of terms that are not divisible by 10. The numbers m = 24 * 10^(294*k - 292) + 1, k = 7*a - 6, a >= 1, are divisible by 7^4 = digsum(m)^4. Also, the numbers s = 491 * 10^(4624*k - 4623) + 3, k = 17*u - 11, u >= 1, are divisible by 17^4 = digsum(s)^4. - Marius A. Burtea, Mar 19 2020

The numbers 2^A095412(n), n >= 6, are terms. - Marius A. Burtea, Apr 02 2020

			k=614656: sumdigits(614656)=28, q=1, since k=28*28*28*28.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A005349, A003634, A072081, A072082.

[k:k in [1..640000]| k mod &+Intseq(k)^4 eq 0]; // Marius A. Burtea, Mar 19 2020

sud[x_] := Apply[Plus, IntegerDigits[x]] Do[s=sud[n]^4; If[IntegerQ[n/s], Print[n]], {n, 1, 10000}]
Select[Range[700000],Divisible[#,Total[IntegerDigits[ #]]^4]&] (* Harvey P. Dale, Jun 28 2011 *)

isok(m) = (m % sumdigits(m)^4) == 0; \\ Michel Marcus, Apr 02 2020

cp's OEIS Frontend

A007471 Sum of digits of n a(n) is n ( = A003634/n), or 0 if no such number exists.

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A057147 a(n) = n times sum of digits of n.

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A003635 Inconsummate numbers in base 10: no number is this multiple of the sum of its digits (in base 10).

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A052489 Largest number that is n times sum of its decimal digits.

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A072081 Numbers divisible by the square of the sum of their digits in base 10.

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Magma

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A056770 Smallest number that is n times the product of its digits or 0 if impossible.

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A065879 a(n) is the smallest positive number that is n times the number of 1's in its binary expansion, or 0 if no such number exists.

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Mathematica

A037478 Number of positive solutions to "numbers that are n times sum of their digits".

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