A005349 -id:A005349 - cp's OEIS frontend

A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 200, 300, 400, 500, 600, 700, 800, 900, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 1000, 1009, 1018, 1027, 1036, 1045, 1054, 1063

Offset: 1

Patrick De Geest, Nov 15 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A052019, A052020, A052021, A052022, A007953, A005349, A028834.
Cf. A175688. - Reinhard Zumkeller, Aug 15 2010
Cf. A055642, A004426.
Cf. A119246, A138166.

import Data.List (isInfixOf)
a052018 n = a052018_list !! (n-1)
a052018_list = filter f [0..] where
   f x = show (a007953 x) `isInfixOf` show x
-- Reinhard Zumkeller, Jun 18 2013

sdssQ[n_]:=Module[{idn=IntegerDigits[n],s,len},s=Total[idn];len= IntegerLength[ s]; MemberQ[Partition[idn,len,1],IntegerDigits[s]]]; Join[{0},Select[Range[1100],sdssQ]] (* Harvey P. Dale, Jan 02 2013 *)

loop = (str(n) for n in range(399))
print([int(n) for n in loop if str(sum(int(k) for k in n)) in n]) # Jonathan Frech, Jun 05 2017

A064438 Numbers which are divisible by the sum of their quaternary digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42, 48, 50, 52, 54, 60, 63, 64, 66, 68, 69, 72, 76, 78, 80, 81, 84, 88, 90, 91, 96, 100, 102, 108, 112, 114, 120, 126, 128, 129, 132, 136, 138, 140, 144, 148, 150, 154, 156, 160, 162, 168, 171, 180

Offset: 1

Len Smiley, Oct 01 2001

A good "puzzle" sequence -- guess the rule given the first twenty or so terms.

			Quaternary representation of 28 is 130, 1 + 3 + 0 = 4 divides 28.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.

Cf. A005349 (decimal), A049445 (binary), A064150 (ternary).

maxarg := 190; for n := 1 to maxarg do if n mod sum(quaternarray(n)) = 0 then write(n," "); end; end; function quaternarray(n: integer): array; var k: integer; stk: stack; begin while n > 0 do k := n mod 4; stack_push(stk,k); n := (n - k) div 4; end; return stack2array(stk); end;

Select[Range[200],Divisible[#,Total[IntegerDigits[#,4]]]&] (* Harvey P. Dale, Jun 09 2011 *)

isok(n) = !(n % sumdigits(n, 4)); \\ Michel Marcus, Jun 24 2018

from sympy.ntheory.factor_ import digits
print([n for n in range(1, 201) if n%sum(digits(n, 4)[1:]) == 0]) # Indranil Ghosh, Apr 24 2017

More terms from Matthew Conroy, Oct 02 2001

Offset changed from 0 to 1 by Harry J. Smith, Sep 14 2009

A069537 Multiples of 2 whose digit sum is 2.

Original entry on oeis.org

2, 20, 110, 200, 1010, 1100, 2000, 10010, 10100, 11000, 20000, 100010, 100100, 101000, 110000, 200000, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000, 100000010, 100000100, 100001000

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002024, A002260, A088404 (half).
Cf. A069521 to A069530, A069532, A069534, A069536.
Subsequence of A005349.
Row n=2 of A245062.

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=s+(r>s), y=if(r>s,r-s,r+s)>>1); 10^x + 10^y; \\ Kevin Ryde, Jul 17 2025

from itertools import product
def agen():
  digits = 1
  while True:
    for i in range(digits-2): yield int("1"+"0"*(digits-3-i)+"1"+"0"*i+"0")
    yield int("2"+"0"*(digits-1))
    digits += 1
g = agen()
print([next(g) for i in range(32)]) # Michael S. Branicky, Feb 20 2021

a(n) = 10^A002024(n-1) + 10^A002260(n-1) for n >= 2. - Kevin Ryde, Jul 17 2025

Corrected and extended by Ray Chandler, Sep 28 2003

A245062 Array read by upward antidiagonals: Niven (or Harshad) numbers arranged in rows by their digit sums.

Original entry on oeis.org

1, 2, 10, 3, 20, 100, 4, 12, 110, 1000, 5, 40, 21, 200, 10000, 6, 50, 112, 30, 1010, 100000, 7, 24, 140, 220, 102, 1100, 1000000, 8, 70, 42, 230, 400, 111, 2000, 10000000, 9, 80, 133, 60, 320, 1012, 120, 10010, 100000000, 190, 18, 152, 322, 114, 410, 1120, 201, 10100, 1000000000

Offset: 1

L. Edson Jeffery, Jul 10 2014

The n-th row contains in increasing order all multiples of n with digit sum n.

See A005349 for definitions and references.

			Array begins as:
  1  10  100  1000  10000  100000  1000000  10000000  100000000  1000000000
  2  20  110   200   1010    1100     2000     10010      10100       11000
  3  12   21    30    102     111      120       201        210         300
  4  40  112   220    400    1012     1120      1300       2020        2200
  5  50  140   230    320     410      500      1040       1130        1220
  6  24   42    60    114     132      150       204        222         240
  7  70  133   322    511     700     1015      1141       1204        1330
  8  80  152   224    440     512      800      1016       1160        1232
  9  18   27    36     45      54       63        72         81          90
190 280  370   460    550     640      730       820        910        1090

Alois P. Heinz, Antidiagonals n = 1..70, flattened
Eric W. Weisstein, Harshad Number, from MathWorld.

Cf. A002998 (column 1), A245065 (column 2).
Cf. A011557 (row 1), A069537 (row 2), A052217 (row 3), A063997 (row 4), A069540 (row 5), A062768 (row 6), A063416 (row 7), A069543 (row 8), A052223 (row 9).
Cf. A082260 (main diagonal).
Cf. A007953, A005349 (Niven or Harshad numbers).
Cf. A082259.

A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 12, 13, 14, 18, 24, 26, 27, 30, 34, 36, 48, 55, 60, 64, 68, 69, 72, 78, 84, 86, 87, 88, 89, 90, 93, 94, 96, 99, 100, 102, 108, 110, 112, 116, 120, 140, 144, 150, 155, 156, 160, 172, 176, 177, 178, 180, 183, 184, 188, 192, 195, 196, 200, 204

Offset: 1

Amiram Eldar, Jan 08 2020

The k-th Fibonacci number is a term for all odd k, since its negaFibonacci representation is 1 followed by (k-1) zeros.

			4 is a term since the negaFibonacci representation of 4 is 10010 whose sum of digits is 1 + 0 + 0 + 1 + 0 = 2 which is a divisor of 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005349, A215022, A328208, A328212, A331083.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]]; f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i]; negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s]; Select[Range[200], Divisible[#, negaFibTermsNum[#]] &]

A001101 Moran numbers: k such that k/(sum of digits of k) is prime.

Original entry on oeis.org

18, 21, 27, 42, 45, 63, 84, 111, 114, 117, 133, 152, 153, 156, 171, 190, 195, 198, 201, 207, 209, 222, 228, 247, 261, 266, 285, 333, 370, 372, 399, 402, 407, 423, 444, 465, 481, 511, 516, 518, 531, 555, 558, 592, 603

Offset: 1

Bill Moran (moran1(AT)llnl.gov)

Witno conjectures that a(n) ~ c*n log(n)^2 for some c. - Charles R Greathouse IV, Jul 26 2011

Bill Moran, Problem 2074: The Moran Numbers, J. Rec. Math., Vol. 25 No. 3, pp. 215, 1993.

Aaron Toponce, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Amin Witno, Numbers which factor as their digital sum times a prime, International Journal of Open Problems in Computer Science and Mathematics 3:2 (2010), pp. 132-136.

Subsequence of A005349, Niven (or Harshad) numbers.

Cf. A007953, A010051, A085775, A108780, A130338, A062339.

import Data.List (findIndices)
a001101 n = a001101_list !! (n-1)
a001101_list = map succ $ findIndices p [1..] where
   p n = m == 0 && a010051 n' == 1 where
      (n', m) = divMod n (a007953 n)
-- Reinhard Zumkeller, Jun 16 2011

filter:= proc(n) local q;
  q:= n/convert(convert(n,base,10),`+`);
  q::integer and isprime(q)
end proc:
select(filter, [$1..1000]); # Robert Israel, May 13 2025

Select[Range[700], PrimeQ[ # / Total[IntegerDigits[#]]]&] (* Jean-François Alcover, Nov 30 2011 *)

is(n)=(k->denominator(k)==1&&isprime(k))(n/sumdigits(n)) \\ Charles R Greathouse IV, Jan 10 2014

from sympy import isprime
def ok(n): s = sum(map(int, str(n))); return s and n%s==0 and isprime(n//s)
print([k for k in range(604) if ok(k)]) # Michael S. Branicky, Mar 28 2022

Name corrected by Charles R Greathouse IV, Jan 10 2014

A106039 Belgian-0 numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 20, 21, 22, 24, 26, 27, 30, 31, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 53, 54, 55, 60, 62, 63, 66, 70, 71, 72, 77, 80, 81, 84, 88, 90, 93, 99, 100, 101, 102, 106, 108, 110, 111, 112, 114, 117, 120

Offset: 1

Eric Angelini, Jun 07 2005

Given an integer -1 < k < 10, n is a Belgian-k number if an infinite sequence in ascending order can be constructed starting at k and including n, and the first differences of that sequence give the base 10 digits of n repeatedly and no others.
Mauro Fiorentini (see Angelini link) explains that all base 10 Harshad numbers (A005349) are also Belgian-0 numbers. - Alonso del Arte, Feb 13 2014
A257778(a(n)) = A257770(a(n),0) = 0. - Reinhard Zumkeller, May 08 2015
Every integer in this sequence is also a Belgian-k number, where k is the sum of digits of the integer. - Davide Rotondo, Jun 12 2024

			13 is a Belgian-0 number because of the sequence
0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, ...
the first differences of which are
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
176 is a Belgian-0 number because, starting from 0 (the seed), one can build a sequence containing 176 in this way:
0.1.8.14.15.22.28.29.36.42.43.50.....155.162.168.169.176.... (sequence)
.1.7.6..1..7..6..1..7..6..1..7..........7...6...1...7.. (first differences)
14 is not a Belgian number because, although we can construct a sequence with the required starting point and the required first differences (namely 0, 1, 5, 6, 10, 11, 15, ...), that sequence does not contain 14.

Vincenzo Librandi and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from Vincenzo Librandi
Eric Angelini, Belgian numbers.
Eric Angelini, Belgian Numbers [Cached copy with permission]

Cf. A257782 (complement), A253717 (primes).
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770, A257778.

a106039 n = a106039_list !! (n-1)
a106039_list = filter belge0 [0..] where
   belge0 n = n == (head $ dropWhile (< n) $
                    scanl (+) 0 $ cycle ((map (read . return) . show) n))
-- Reinhard Zumkeller, May 07 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 120, belgianQ[#, 0] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A341169 Numbers that divided by the sum of their digits leave 2 as remainder.

Original entry on oeis.org

16, 22, 26, 32, 67, 82, 86, 122, 130, 142, 170, 172, 178, 184, 202, 205, 212, 242, 262, 302, 310, 314, 331, 338, 352, 359, 418, 442, 464, 466, 496, 520, 530, 532, 535, 590, 602, 622, 652, 665, 667, 712, 716, 754, 802, 818, 838, 842, 968, 971, 1022, 1024, 1030, 1034, 1072, 1091, 1094

Offset: 1

Eric Angelini and Carole Dubois, Feb 06 2021

			a(1) = 16 and 16 is 7*2 with remainder 2;
a(2) = 22 and 22 is 4*5 with remainder 2; etc.

_Carole Dubois_, Table of n, a(n) for n = 1..5001

Cf. A005349 (Niven numbers: remainder = 0), A209871 (Quasi-Niven numbers: remainder = 1), A341169 to A341182 (remainders = 2 to 15).

A064481 Numbers which are divisible by the sum of their base-5 digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 26, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 51, 52, 54, 56, 60, 63, 64, 65, 66, 72, 75, 76, 78, 80, 85, 88, 90, 91, 96, 99, 100, 102, 104, 105, 112, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 145

Offset: 1

Klaus Brockhaus, Oct 03 2001

			Base-5 representation of 28 is 103; 1 + 0 + 3 = 4 divides 28.

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

Cf. A005349 (base 10), A049445 (base 2), A064150 (base 3), A064438 (base 4), A344341.

: maxarg := 160; for n := 1 to maxarg do if n mod sum(basearray(n,5)) = 0 then write(n," "); end; end; function basearray(n,b: integer): array; var k: integer; stk: stack; begin while n > 0 do k := n mod b; stack_push(stk,k); n := (n - k) div b; end; return stack2array(stk); end;.

isok(n) = !(n % sumdigits(n, 5)); \\ Michel Marcus, Jun 24 2018

Offset changed from 0 to 1 by Harry J. Smith, Sep 15 2009

A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

Original entry on oeis.org

1, 2, 6, 12, 15, 16, 18, 20, 30, 35, 36, 45, 48, 55, 60, 70, 72, 78, 84, 90, 91, 95, 96, 98, 104, 108, 132, 144, 147, 154, 168, 175, 184, 189, 208, 224, 231, 232, 245, 252, 256, 261, 264, 270, 275, 280, 282, 287, 294, 315, 322, 324, 330, 336, 340, 342, 351, 357

Offset: 1

Amiram Eldar, Apr 22 2020

			6 is a term since its base phi representation is 1010.0001, and the number of 1's is 3, which is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Wikipedia, Golden ratio base.

Cf. A005349, A049445, A055778, A064150, A064438, A064481, A118363, A328208, A328212, A333426.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]]; Select[Range[360], Divisible[#, phiDigSum[#]] &]

cp's OEIS Frontend

A052018 Numbers k with the property that the sum of the digits of k is a substring of k.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A064438 Numbers which are divisible by the sum of their quaternary digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

Mathematica

PARI

Python

Extensions

A069537 Multiples of 2 whose digit sum is 2.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A245062 Array read by upward antidiagonals: Niven (or Harshad) numbers arranged in rows by their digit sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A331085 Positive negaFibonacci-Niven numbers: positive numbers divisible by the number of terms in their negaFibonacci representation (A331083).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A001101 Moran numbers: k such that k/(sum of digits of k) is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

A106039 Belgian-0 numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica