This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A368106 #7 Dec 12 2023 08:02:05
%S A368106 1,1,1,2,1,1,1,4,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,4,2,1,4,2,1,1,1,4,1,1,
%T A368106 1,4,1,1,1,4,1,1,1,2,2,1,1,2,2,2,1,2,1,4,1,4,1,1,1,2,1,1,2,4,1,1,1,2,
%U A368106 1,1,1,8,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N A368106 The number of infinitary divisors of the powerful part of n.
%H A368106 Amiram Eldar, <a href="/A368106/b368106.txt">Table of n, a(n) for n = 1..10000</a>
%F A368106 a(n) = A037445(A057521(n)).
%F A368106 Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.
%F A368106 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A368106 a(n) <= A037445(n), with equality if and only if n is powerful (A001694).
%F A368106 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.89684906463124350536..., where f(x) = (1-x) * (Product_{k>=0} (1 + 2*x^(2^k)) - x).
%t A368106 f[p_, e_] := If[e == 1, 1, 2^DigitCount[e, 2, 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368106 (PARI) a(n) = vecprod(apply(x -> if(x == 1, 1, 2^hammingweight(x)), factor(n)[, 2]));
%Y A368106 Cf. A001694, A005117, A037445, A057521.
%Y A368106 Similar sequences: A323308, A357669, A368104.
%K A368106 nonn,easy,mult
%O A368106 1,4
%A A368106 _Amiram Eldar_, Dec 12 2023