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

1,2

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

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]

Prog

(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); }

Crossrefs

Subsequence of A387056.

Cf. A037445, A077609.

Sequence in context: A099418 A108987 A035405 * A280867 A111058 A063622

Adjacent sequences: A387054 A387055 A387056 * A387058 A387059 A387060

Keyword

nonn,easy

Author

Amiram Eldar, Aug 15 2025

Status

approved