cp's OEIS Frontend

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

%I A387057 #7 Aug 17 2025 11:18:03
%S A387057 1,2,8,12,20,24,28,36,40,44,52,56,64,68,72,76,88,92,100,104,116,124,
%T A387057 128,136,148,152,164,172,184,188,196,200,212,232,236,244,248,268,284,
%U A387057 292,296,316,324,328,332,344,356,376,384,388,392,404,412,424,428,436,452
%N A387057 Numbers k that are infinitarily divisible by the number of infinitary divisors of k.
%C A387057 Numbers k such that A037445(k) is an infinitary divisor of k.
%C A387057 This sequence is infinite. For example, if p is an odd prime, then 8*p is a term.
%H A387057 Amiram Eldar, <a href="/A387057/b387057.txt">Table of n, a(n) for n = 1..10000</a>
%t A387057 infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (fct = FactorInteger[d])], (e = Last /@ fct)] == e;
%t A387057 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]
%o A387057 (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
%o A387057 isok(k) = {my(f = factor(k), id = vecprod(apply(x -> 2^hammingweight(x), f[, 2]))); !(k % id) && isidiv(id, f);}
%Y A387057 Subsequence of A387056.
%Y A387057 Cf. A037445, A077609.
%K A387057 nonn,easy
%O A387057 1,2
%A A387057 _Amiram Eldar_, Aug 15 2025