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

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

Author

Bernard Schott, Feb 20 2022

Keywords

• nonn
• base

Comments

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).

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Python

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

Extensions

More terms from Amiram Eldar, Feb 21 2022