A387057 Numbers k that are infinitarily divisible by the number of infinitary divisors of k.

1, 2, 8, 12, 20, 24, 28, 36, 40, 44, 52, 56, 64, 68, 72, 76, 88, 92, 100, 104, 116, 124, 128, 136, 148, 152, 164, 172, 184, 188, 196, 200, 212, 232, 236, 244, 248, 268, 284, 292, 296, 316, 324, 328, 332, 344, 356, 376, 384, 388, 392, 404, 412, 424, 428, 436, 452

Offset: 1

Author

Amiram Eldar, Aug 15 2025

Keywords

• nonn
• easy

Comments

Numbers k such that A037445(k) is an infinitary divisor of k.

This sequence is infinite. For example, if p is an odd prime, then 8*p is a term.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A387056.

Cf. A037445, A077609.

Programs

Mathematica

infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (fct = FactorInteger[d])], (e = Last /@ fct)] == e;
f[p_, e_] := 2^DigitCount[e, 2, 1]; id[1] = 1; id[n_] := Times @@ f @@@ FactorInteger[n]; q[k_] := Module[{d = id[k]}, Divisible[k, d] && infDivQ[k, d]]; Select[Range[500], q]

PARI

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); } \\ Michel Marcus at A077609
isok(k) = {my(f = factor(k), id = vecprod(apply(x -> 2^hammingweight(x), f[, 2]))); !(k % id) && isidiv(id, f);}