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
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.
Programs
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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
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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
Comments