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.

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

Author

Bernard Schott, Feb 24 2022

Keywords

• nonn
• base

Comments

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.

Examples

			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.
		

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Extensions

More terms from Amiram Eldar, Feb 24 2022