A351809 -id:A351809 - 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

A351808 a(n) is the quotient obtained when pod(m) divides pod(m^2), with pod = product of digits = A007954 and m = A351807(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 2, 8, 18, 4, 10, 3, 2, 8, 32, 15, 6, 21, 9, 14, 18, 0, 0, 6, 12, 21, 24, 0, 0, 0, 0, 0, 0, 0, 0, 7, 32, 81, 0, 10, 4, 0, 30, 35, 0, 21, 144, 0, 64, 32, 0, 2, 0, 0, 0, 12, 80, 252, 243, 12, 60, 27, 48, 256, 15, 30, 140, 36, 8, 14, 336, 96, 144

Offset: 1

Bernard Schott, Feb 20 2022

a(n) = 0 iff m = A351807(n) is a term of A134844.

As pod(m) is 7-smooth number and pod(m^2) can be 0 (see example), all terms of the sequence are in {0} union A002473. The smallest term k such that the corresponding quotient = 0 or A002473(n) is A351809(n).

			A351807(9) = 13, then pod(13) = 1*3 = 3 while pod(13^2) = pod(169) = 1*6*9 = 54; hence, a(9) = 54/3 = 18.
A351807(23) = 33, then pod(33) = 3*3 = 9 while pod(33^2) = pod(1089) = 1*0*8*9 = 0; hence, a(23) = 0.

Cf. A002473, A007954, A134844, A351807, A351809.

pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, 1/2]; Select[r /@ Range[200], IntegerQ] (* Amiram Eldar, Feb 21 2022 *)

lista(nn) = {my(list=List()); for (m=1, nn, my(d=digits(m), q); if (vecmin(d) && denominator(q = vecprod(digits(m^2))/vecprod(d)) == 1, listput(list, q);); ); Vec(list);} \\ Michel Marcus, Feb 21 2022

from math import prod
from itertools import count, islice
def A351808_gen(): # generator of terms
    return (q for q, r in (divmod(prod(int(d) for d in str(m**2)),prod(int(d) for d in str(m))) for m in count(1) if '0' not in str(m)) if r == 0)
A351808_list = list(islice(A351808_gen(),20)) # Chai Wah Wu, Feb 25 2022

More terms from Amiram Eldar, Feb 21 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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