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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 12 01:50 EDT 2021. Contains 343808 sequences. (Running on oeis4.)