A003635 -id:A003635 - cp's OEIS frontend

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

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

A058898 Inconsummate numbers in base 2: no number is this multiple of the sum of its digits (in base 2).

Original entry on oeis.org

13, 19, 25, 26, 35, 38, 47, 49, 50, 52, 55, 67, 70, 76, 94, 95, 97, 98, 100, 103, 104, 109, 110, 115, 117, 131, 134, 140, 151, 152, 157, 159, 171, 175, 179, 183, 185, 187, 188, 190, 193, 194, 196, 199, 200, 203, 206, 208, 217, 218, 220, 227, 229

Offset: 1

N. J. A. Sloane, Jan 09 2001

Equivalently, these are the natural numbers that cannot be written as the arithmetic mean of distinct powers of 2. - Brian Kell, Feb 28 2009

Daniel Mondot, Table of n, a(n) for n = 1..33099

Cf. A003635, A052491, A058899-A058907.

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

Do[k = n; While[ Apply[ Plus, IntegerDigits[k, 2] ]*n != k && k < 250n, k += n]; If[k == 250n, Print[n] ], {n, 1, 10^3} ]

from itertools import count, islice, combinations_with_replacement
def A058898_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if l*n < 1<0 and sorted(bin(s*n)[2:]) == [str(e) for e in d]:
                    break
            else:
                continue
            break
A058898_list = list(islice(A058898_gen(),20)) # Chai Wah Wu, May 09 2023

n such that A065413(n) = 0. - Brian Kell, Mar 01 2009

A052491 Smallest "inconsummate number" in base n: smallest number such that in base n, no number is this multiple of the sum of its digits.

Original entry on oeis.org

13, 17, 29, 16, 27, 30, 42, 46, 62, 68, 86, 92, 114, 122, 147, 154, 182, 192, 222, 232, 266, 278, 314, 326, 367, 380, 422, 436, 482, 498, 546, 562, 614, 632, 688, 704, 762, 782, 842, 862, 926, 948, 1014, 1036, 1107, 1130, 1202, 1226, 1302, 1328, 1406, 1432

Offset: 2

J. H. Conway, Dec 30 2000

			a(10) = 62, from A003635.

Chai Wah Wu, Table of n, a(n) for n = 2..174

Cf. A003635, A058898-A058907.

Do[n = 1; While[k = n; While[ Apply[ Plus, IntegerDigits[k, b] ]*n != k && k < 100n, k += n ]; k != 100n, n++ ]; Print[n], {b, 2, 54} ]

from itertools import count, combinations_with_replacement
from sympy.ntheory import digits
def A052491(n):
    for k in count(1):
        for l in count(1):
            if (n-1)*l*k < n**(l-1):
                return k
            for d in combinations_with_replacement(range(n),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*k,n)[1:]) == list(d):
                    break
            else:
                continue
            break # Chai Wah Wu, May 09 2023

Corrected and extended by David Radcliffe, Jan 08 2001

More terms from David W. Wilson, Jan 10 2001

A003634 Smallest positive integer that is n times its digit sum, or 0 if no such number exists.

Original entry on oeis.org

1, 18, 27, 12, 45, 54, 21, 72, 81, 10, 198, 108, 117, 126, 135, 144, 153, 162, 114, 180, 378, 132, 207, 216, 150, 234, 243, 112, 261, 270, 372, 576, 594, 102, 315, 324, 111, 342, 351, 120, 738, 756, 516, 792, 405, 230, 423, 432, 441, 450, 918, 312, 954, 972

Offset: 1

N. J. A. Sloane, Mira Bernstein

a(n) = 0 for n = 62, 63, 65, ... (A003635). - Robert G. Wilson v, Aug 15 2000

			a(3) = 27 because no number less than 27 has a digit sum equal to 3 times the number.

J. H. Conway, personal communication.
Anthony Gardiner, Mathematical Puzzling, Dover Publications, Inc., Mineola, NY, 1987, Page 11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A005349, A003635.

Do[k = n; While[Apply[Plus, RealDigits[k][[1]]]*n != k, k += n]; Print[k], {n, 1, 61}]
With[{ll=Select[Table[{n,n/Total[IntegerDigits[n]]},{n,1000}],IntegerQ[ #[[2]]]&]},Table[Select[ll,#[[2]]==i&,1][[1,1]],{i,60}]] (* Harvey P. Dale, Mar 09 2012 *)

def sd(n): return sum(map(int, str(n)))
def a(n):
  m = 1
  while m != n*sd(m): m += 1
  return m
print([a(n) for n in range(1,62)]) # Michael S. Branicky, Jan 18 2021

from itertools import count, combinations_with_replacement
def A003634(n):
    for l in count(1):
        if 9*l*n < 10**(l-1): return 0
        c = 10**l
        for d in combinations_with_replacement(range(10),l):
            if sorted(str(a:=sum(d)*n)) == [str(e) for e in d] and a>0:
                c = min(c,a)
        if c < 10**l:
            return c # 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

A058907 Inconsummate numbers in base 12: no number is this multiple of the sum of its digits (in base 12).

Original entry on oeis.org

86, 87, 88, 90, 99, 101, 102, 112, 113, 114, 125, 126, 138, 229, 235, 244, 245, 246, 256, 258, 269, 270, 282, 307, 373, 379, 385, 391, 392, 400, 402, 426, 451, 464, 530, 535, 536, 542, 543, 547, 548, 607, 608, 620, 667, 673, 674, 679, 680, 685

Offset: 1

N. J. A. Sloane, Jan 09 2001

Vincenzo Librandi, Table of n, a(n) for n = 1..10100

Cf. A003635, A052491, A058898, A058899, A058900, A058901, A058902, A058903, A058904, A058905, A058906.

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

base=12; Do[k=n; While[Apply[Plus, IntegerDigits[k, base]] n!=k&&k<250n, k+=n]; If[k==250 n, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Sep 23 2017; after N. J. A. Sloane in A058906 *)

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058907_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 11*l*n < 12**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(12),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,12)[1:]) == list(d):
                    break
            else:
                continue
            break
A058907_list = list(islice(A058907_gen(),20)) # Chai Wah Wu, May 10 2023

A065877 Non-Niven (or non-Harshad) numbers: numbers which are not a multiple of the sum of their digits.

Original entry on oeis.org

11, 13, 14, 15, 16, 17, 19, 22, 23, 25, 26, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101

Offset: 1

Henry Bottomley, Nov 26 2001

A188641(a(n)) = 0; A070635(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2011

			13 is in the list because 13 is not a multiple of 1 + 3 = 4.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Complement of A005349. Cf. A003635, A007953, A065878.

Cf. A188643.

import Data.List (findIndices)
a065877 n = a065877_list !! (n-1)
a065877_list = map succ $ findIndices (> 0) $ map a070635 [1..]
-- Reinhard Zumkeller, Apr 07 2011

Select[Range[101],!IntegerQ[#/Total[IntegerDigits[#]]] &] (* Jayanta Basu, May 05 2013 *)

isok(k) = {(k % sumdigits(k)) != 0} \\ Harry J. Smith, Nov 03 2009

A058906 Inconsummate numbers in base 11: no number is this multiple of the sum of its digits (in base 11).

Original entry on oeis.org

68, 70, 79, 80, 82, 92, 104, 200, 202, 212, 214, 224, 225, 248, 260, 314, 320, 332, 380, 392, 452, 458, 464, 490, 502, 508, 512, 513, 514, 518, 520, 524, 530, 562, 568, 574, 578, 579, 580, 584, 585, 590, 592, 595, 598, 599, 622, 628, 634, 635

Offset: 1

N. J. A. Sloane, Jan 09 2001

Cf. A003635, A052491, A058898-A058907.

digitsum := proc (n,b) local i; add(i,i=convert(n,base,b)) end; b := 11:N := 43922; L := []: for n from 1 to N do k := digitsum(n,b): if (n mod k)=0 then L := [op(L), n/k] fi: od: M := []: for i from 1 to 1000 do if not(member(i,L)) then M := [op(M),i] fi od: lprint(M);

base = 11; Do[k = n; While[ Apply[ Plus, IntegerDigits[k, base] ]*n != k && k < 250n, k += n]; If[k == 250n, Print[n] ], {n, 1, 10^3} ]

from itertools import count, islice, combinations_with_replacement
from sympy.ntheory import digits
def A058906_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        for l in count(1):
            if 10*l*n < 11**(l-1):
                yield n
                break
            for d in combinations_with_replacement(range(11),l):
                if (s:=sum(d)) > 0 and sorted(digits(s*n,11)[1:]) == list(d):
                    break
            else:
                continue
            break
A058906_list = list(islice(A058906_gen(),20)) # Chai Wah Wu, May 09 2023

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.

cp's OEIS Frontend

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

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A058898 Inconsummate numbers in base 2: no number is this multiple of the sum of its digits (in base 2).

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A052491 Smallest "inconsummate number" in base n: smallest number such that in base n, no number is this multiple of the sum of its digits.

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A003634 Smallest positive integer that is n times its digit sum, or 0 if no such number exists.

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

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A058907 Inconsummate numbers in base 12: no number is this multiple of the sum of its digits (in base 12).

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Maple

Mathematica

Python

A065877 Non-Niven (or non-Harshad) numbers: numbers which are not a multiple of the sum of their digits.

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