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

Author

Bernard Schott, Jan 15 2021

Keywords

• nonn
• base

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)).

Examples

			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.
		

Crossrefs

Cf. A007954, A069035, A087304.

Programs

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 *)

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

Formula

a(10*k) = 20*k.

Extensions

More terms from Amiram Eldar, Jan 15 2021