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A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.

%I A365549 #15 Apr 27 2025 00:45:29
%S A365549 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T A365549 1,4,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,3,1,1,1,2,
%U A365549 1,1,1,4,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N A365549 The number of exponentially odd divisors of the square root of the largest square dividing n.
%C A365549 First differs from A278908, A307848, A323308 and A358260 at n = 64.
%C A365549 The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n).
%H A365549 Amiram Eldar, <a href="/A365549/b365549.txt">Table of n, a(n) for n = 1..10000</a>
%F A365549 a(n) = A322483(A000188(n)).
%F A365549 a(n) >= 1 with equality if and only if n is squarefree (A005117).
%F A365549 Multiplicative with a(p^e) = 2 + floor((e-2)/4).
%F A365549 Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 + 1/p^(2*s) - 1/p^(4*s)).
%F A365549 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.54211628314015874165... .
%t A365549 f[p_, e_] := 2 + Floor[(e-2)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365549 (PARI) a(n) = vecprod(apply(x -> 2 + (x-2)\4, factor(n)[, 2]));
%Y A365549 Cf. A000188, A005117, A013662, A046951, A322483.
%Y A365549 Cf. A278908, A307848, A323308, A358260.
%K A365549 nonn,easy,mult
%O A365549 1,4
%A A365549 _Amiram Eldar_, Sep 08 2023