A351808 -id:A351808 - cp's OEIS frontend

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

Bernard Schott, Feb 19 2022

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).

			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.

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

Subsequences: A002275, A134844.

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

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

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

More terms from Amiram Eldar, Feb 19 2022

A351809 a(0) = 32; then, for n >= 1, a(n) is the smallest positive integer k such that pod(k^2)/pod(k) = A002473(n) where pod = product of digits = A007954.

Original entry on oeis.org

32, 1, 2, 3, 15, 381, 25, 61, 12, 27, 16, 41, 28, 23, 336, 13, 1766, 26, 43, 2675, 118, 278, 74, 22, 76, 128, 392, 343, 228, 121, 418, 976, 258, 193, 116, 194, 93, 218, 441, 1231, 112, 63, 219, 984, 136, 4165, 2271, 1894, 183, 615, 434, 22831, 523, 1592, 2435

Offset: 0

Bernard Schott, Feb 24 2022

As pod(m) is a 7-smooth number and pod(m^2) can be 0, all terms of A351808 are in {0} union A002473. See example section for why a(0) = 32.

			pod(32) = 3*2 = 6, pod(32^2) = pod(1024) = 1*0*2*4 = 0, and k = 32 is the smallest positive integer k such that pod(k^2) = 0 while pod(k) <> 0, so a(0) = 32.
A002473(5) = 5; pod(381) = 3*8*1 = 24, pod(381^2) = pod(145161) = 1*4*5*1*6*1 = 120; as 120/24 = 5, and 381 is the smallest positive integer k such that pod(k^2)/pod(k) = 5 then a(5) = 381.
A002473(11) = 12; pod(41)= 4*1 = 4, pod(41^2) = pod(1681) = 1*6*8*1 = 48; as 48/4 = 12 and 41 is the smallest positive integer k such that pod(k^2)/pod(k) = 12, then a(11) = 41.

Cf. A002473, A007954, A280012, A351807, A351808.

sevenSmooths = Select[Range[150], Max[FactorInteger[#][[;; , 1]]] <= 7 &]; pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, -1]; s = Array[r, 3*10^4]; TakeWhile[FirstPosition[s, #] & /@ Join[{0}, sevenSmooths] // Flatten, NumberQ] (* Amiram Eldar, Feb 24 2022 *)

pod(k) = vecprod(digits(k)); \\ A007954
smp(m) = my(k=1); while (!pod(k) || (pod(k^2)/pod(k) != m), k++); k;
isss(n) = (n<11) || (vecmax(factor(n, 7)[, 1])<8); \\ A002473
lista(nn) = apply(smp, select(isss, [0..nn]));
lista(200) \\ Michel Marcus, Feb 24 2022

More terms from Amiram Eldar, Feb 24 2022

cp's OEIS Frontend

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

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