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.

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

1,1

Comments

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)).

Links

Table of n, a(n) for n=1..43.

Formula

a(10*k) = 20*k.

Example

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.

Mathematica

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 *)

Prog

(PARI) 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
(Python)
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

Crossrefs

Cf. A007954, A069035, A087304.

Sequence in context: A056684 A042731 A087304 * A121808 A160265 A007651

Adjacent sequences: A340201 A340202 A340203 * A340205 A340206 A340207

Keyword

nonn,base

Author

Bernard Schott, Jan 15 2021

Extensions

More terms from Amiram Eldar, Jan 15 2021

Status

approved