A069035 -id:A069035 - cp's OEIS frontend

A077754 Smallest multiple of n with two or more digits, none of them zeros, whose digit sum equals n, or 0 if no such multiple exists.

0, 0, 12, 112, 0, 24, 133, 152, 18, 0, 0, 48, 247, 266, 195, 448, 476, 198, 874, 0, 399, 2398, 1679, 888, 4975, 1898, 999, 7588, 4988, 0, 8959, 17888, 42999, 28798, 57995, 29988, 37999, 59888, 49998, 0, 177899, 88998, 99889, 479996, 499995, 589996

Offset: 1

Amarnath Murthy, Nov 20 2002

Equivalently, the digits of a(n) are a nontrivial composition of n and n divides a(n).

Conjecture: zero occurs only for indices which are multiples of 10 apart from 1, 2, 5 and 11. (a(n) = 0 only for n = 1,2,5,11 or n = 10k.)

			a(8) = 152 is a multiple of 8; a(10) = 0, since every multiple of 10 includes a 0.

Chai Wah Wu, Table of n, a(n) for n = 1..100

Cf. A002998, A069035.

Edited and extended by Franklin T. Adams-Watters, Jun 14 2006

a(44) corrected by Chai Wah Wu, Mar 17 2016

A269332 Smallest multiple of n whose sum of digits is greater than n.

Original entry on oeis.org

2, 4, 6, 8, 15, 18, 28, 48, 99, 290, 66, 96, 78, 98, 495, 288, 289, 1998, 399, 4980, 798, 968, 897, 3888, 5975, 3978, 19899, 6888, 6699, 69990, 7998, 9888, 9999, 29988, 68985, 299988, 74999, 69996, 79989, 799880, 99999, 298998, 89999, 589996, 4999995, 599978, 498999

Offset: 1

Franklin T. Adams-Watters, Feb 24 2016

Cf. A269333, A007953, A069035.

a(n) = {my(m=2*n); while (sumdigits(m) <= n, m+=n); m;} \\ Michel Marcus, Feb 24 2016

More terms from Michel Marcus, Feb 24 2016

A340204 a(n) is the smallest proper multiple of n whose digit product is the same as the digit product of n; 0 if no such number exists.

Original entry on oeis.org

11, 12, 1113, 212, 15, 132, 11711, 24, 11133, 20, 1111, 11112, 1131, 21112, 11115, 32, 71111, 11124, 133, 40, 11111121, 1122, 161, 14112, 125, 1612, 11111111172, 224, 3132, 60, 11111113, 1312, 11111133, 612, 315, 1332, 11137, 342, 11193, 80, 1111141, 11214, 11223

Offset: 1

Bernard Schott, Jan 15 2021

Every odd integer k not ending with 5 has a multiple that is a repunit (see A099679), hence a(n) <= the concatenation of this repunit with this odd number (see example a(33)).

			a(16) = 32 because 32 is the smallest proper multiple of 16 such that 1*6 = 3*2.
a(33) = 11111133 is the concatenation of 111111 (that is the smallest repunit multiple of 33) with 33.

Cf. A007954, A069035, A087304.

prodig[n_] := Times @@ IntegerDigits[n]; a[n_] := Module[{k = 2*n, p = prodig[n]}, While[prodig[k] != p, k += n]; k]; Array[a, 20] (* Amiram Eldar, Jan 15 2021 *)

f(n) = vecprod(digits(n)); \\ A007954
a(n) = my(x = f(n), k = 2); while(f(k*n) != x, k++); k*n; \\ Michel Marcus, Jan 15 2021

from math import prod
def pd(n): return prod(map(int, str(n)))
def a(n):
  pdn, f = pd(n), 2
  while pd(f*n) != pdn: f += 1
  return f*n
print([a(n) for n in range(1, 27)]) # Michael S. Branicky, Jan 16 2021

a(10*k) = 20*k.

More terms from Amiram Eldar, Jan 15 2021

cp's OEIS Frontend

A077754 Smallest multiple of n with two or more digits, none of them zeros, whose digit sum equals n, or 0 if no such multiple exists.

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A269332 Smallest multiple of n whose sum of digits is greater than n.

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A340204 a(n) is the smallest proper multiple of n whose digit product is the same as the digit product of n; 0 if no such number exists.

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