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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.

%I A351809 #20 Feb 28 2022 15:12:25
%S A351809 32,1,2,3,15,381,25,61,12,27,16,41,28,23,336,13,1766,26,43,2675,118,
%T A351809 278,74,22,76,128,392,343,228,121,418,976,258,193,116,194,93,218,441,
%U A351809 1231,112,63,219,984,136,4165,2271,1894,183,615,434,22831,523,1592,2435
%N 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.
%C A351809 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.
%e A351809 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.
%e A351809 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.
%e A351809 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.
%t A351809 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 *)
%o A351809 (PARI) pod(k) = vecprod(digits(k)); \\ A007954
%o A351809 smp(m) = my(k=1); while (!pod(k) || (pod(k^2)/pod(k) != m), k++); k;
%o A351809 isss(n) = (n<11) || (vecmax(factor(n, 7)[, 1])<8); \\ A002473
%o A351809 lista(nn) = apply(smp, select(isss, [0..nn]));
%o A351809 lista(200) \\ _Michel Marcus_, Feb 24 2022
%Y A351809 Cf. A002473, A007954, A280012, A351807, A351808.
%K A351809 nonn,base
%O A351809 0,1
%A A351809 _Bernard Schott_, Feb 24 2022
%E A351809 More terms from _Amiram Eldar_, Feb 24 2022