A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

1, 2, 3, 5, 6, 8, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 36, 41, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 61, 62, 63, 64, 66, 68, 71, 74, 76, 78, 82, 83, 84, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113, 114, 115, 116, 118, 121, 122, 123

Offset: 1

Author

Bernard Schott, Feb 19 2022

Keywords

• nonn
• base

Comments

Inspired by A351650 where pod is replaced by sod.

All terms are zeroless (A052382).

Repunits form a subsequence (A002275).

Integers m without 0 and such that m^2 has a 0 form a subsequence (A134844).

The smallest term k such that the corresponding quotient = n is A351809(n).

Examples

			Product of digits of 27 = 2*7 = 14; then 27^2 = 729, product of digits of 729 = 7*2*9 = 81; as 81 divides 729, 27 is a term.
		

Crossrefs

Cf. A007954, A002473, A351808 (corresponding quotients), A351809.

Subsequences: A002275, A134844.

Programs

Mathematica

pod[n_] := Times @@ IntegerDigits[n]; Select[Range[120], FreeQ[IntegerDigits[#], 0] && Divisible[pod[#^2], pod[#]] &] (* Amiram Eldar, Feb 19 2022 *)

PARI

isok(m) = my(d=digits(m)); vecmin(d) && denominator(vecprod(digits(m^2))/vecprod(d)) == 1; \\ Michel Marcus, Feb 19 2022

Python

from math import prod
def pod(n): return prod(map(int, str(n)))
def ok(m): pdm = pod(m); return pdm > 0 and pod(m*m)%pdm == 0
print([m for m in range(124) if ok(m)]) # Michael S. Branicky, Feb 19 2022

Extensions

More terms from Amiram Eldar, Feb 19 2022