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The number of infinitary divisors of the powerful part of n.
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%I #7 Dec 12 2023 08:02:05

%S 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 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 1,1,1,8,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1

%N The number of infinitary divisors of the powerful part of n.

%H Amiram Eldar, <a href="/A368106/b368106.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A037445(A057521(n)).

%F Multiplicative with a(p) = 1 and a(p^e) = 2^A000120(e) for e >= 2.

%F a(n) >= 1, with equality if and only if n is squarefree (A005117).

%F a(n) <= A037445(n), with equality if and only if n is powerful (A001694).

%F 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 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 (PARI) a(n) = vecprod(apply(x -> if(x == 1, 1, 2^hammingweight(x)), factor(n)[, 2]));

%Y Cf. A001694, A005117, A037445, A057521.

%Y Similar sequences: A323308, A357669, A368104.

%K nonn,easy,mult

%O 1,4

%A _Amiram Eldar_, Dec 12 2023