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Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.
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%I #30 Feb 20 2022 17:17:21

%S 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,

%T 42,43,45,47,48,49,51,52,53,55,61,62,63,64,66,68,71,74,76,78,82,83,84,

%U 93,94,95,96,97,98,99,111,112,113,114,115,116,118,121,122,123

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

%C Inspired by A351650 where pod is replaced by sod.

%C All terms are zeroless (A052382).

%C Repunits form a subsequence (A002275).

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

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

%e 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.

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

%o (Python)

%o from math import prod

%o def pod(n): return prod(map(int, str(n)))

%o def ok(m): pdm = pod(m); return pdm > 0 and pod(m*m)%pdm == 0

%o print([m for m in range(124) if ok(m)]) # _Michael S. Branicky_, Feb 19 2022

%o (PARI) isok(m) = my(d=digits(m)); vecmin(d) && denominator(vecprod(digits(m^2))/vecprod(d)) == 1; \\ _Michel Marcus_, Feb 19 2022

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

%Y Subsequences: A002275, A134844.

%K nonn,base

%O 1,2

%A _Bernard Schott_, Feb 19 2022

%E More terms from _Amiram Eldar_, Feb 19 2022