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

%I A351808 #24 Feb 25 2022 09:46:25
%S A351808 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,
%T A351808 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,
%U A351808 252,243,12,60,27,48,256,15,30,140,36,8,14,336,96,144
%N A351808 a(n) is the quotient obtained when pod(m) divides pod(m^2), with pod = product of digits = A007954 and m = A351807(n).
%C A351808 a(n) = 0 iff m = A351807(n) is a term of A134844.
%C A351808 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).
%e A351808 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.
%e A351808 A351807(23) = 33, then pod(33) = 3*3 = 9 while pod(33^2) = pod(1089) = 1*0*8*9 = 0; hence, a(23) = 0.
%t A351808 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 *)
%o A351808 (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
%o A351808 (Python)
%o A351808 from math import prod
%o A351808 from itertools import count, islice
%o A351808 def A351808_gen(): # generator of terms
%o A351808     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)
%o A351808 A351808_list = list(islice(A351808_gen(),20)) # _Chai Wah Wu_, Feb 25 2022
%Y A351808 Cf. A002473, A007954, A134844, A351807, A351809.
%K A351808 nonn,base
%O A351808 1,2
%A A351808 _Bernard Schott_, Feb 20 2022
%E A351808 More terms from _Amiram Eldar_, Feb 21 2022