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A365552 The number of exponentially odd divisors of the powerful part of n.

%I A365552 #9 Sep 09 2023 06:48:16
%S A365552 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,3,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,4,1,1,
%T A365552 1,4,1,1,1,3,1,1,1,2,2,1,1,3,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,4,1,1,1,2,
%U A365552 1,1,1,6,1,1,2,2,1,1,1,3,3,1,1,2,1,1,1
%N A365552 The number of exponentially odd divisors of the powerful part of n.
%C A365552 First differs from A095691 at n = 512.
%H A365552 Amiram Eldar, <a href="/A365552/b365552.txt">Table of n, a(n) for n = 1..10000</a>
%F A365552 a(n) = A322483(A057521(n)).
%F A365552 Multiplicative with a(p) = 1 and a(p^e) = floor((e+3)/2) for e >= 2.
%F A365552 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^(3*s) - 1/p^(4*s)).
%F A365552 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 + 1/p^3 - 1/p^4) = 1.80989829762278336163... .
%t A365552 f[p_, e_] := If[e == 1, 1, Floor[(e + 3)/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365552 (PARI) a(n) = vecprod(apply(x -> if(x == 1, 1, (x+3)\2), factor(n)[, 2]));
%Y A365552 Cf. A013661, A095691, A057521, A322483.
%K A365552 nonn,easy,mult
%O A365552 1,4
%A A365552 _Amiram Eldar_, Sep 08 2023