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A365551 The number of exponentially odd divisors of the smallest exponentially odd number divisible by n.

%I A365551 #12 Sep 09 2023 11:34:01
%S A365551 1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,4,2,6,2,6,4,4,2,6,3,4,3,6,2,8,2,4,4,4,
%T A365551 4,9,2,4,4,6,2,8,2,6,6,4,2,8,3,6,4,6,2,6,4,6,4,4,2,12,2,4,6,5,4,8,2,6,
%U A365551 4,8,2,9,2,4,6,6,4,8,2,8,4,4,2,12,4,4,4
%N A365551 The number of exponentially odd divisors of the smallest exponentially odd number divisible by n.
%C A365551 First differs from A049599 and A282446 at n = 32, and from A353898 at n = 64.
%H A365551 Amiram Eldar, <a href="/A365551/b365551.txt">Table of n, a(n) for n = 1..10000</a>
%H A365551 Vaclav Kotesovec, <a href="/A365551/a365551.jpg">Graph - the asymptotic ratio (100000 terms)</a>.
%F A365551 a(n) = A322483(A356191(n)).
%F A365551 Multiplicative with a(p^e) = ceiling((e+3)/2).
%F A365551 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
%F A365551 From _Vaclav Kotesovec_, Sep 09 2023: (Start)
%F A365551 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
%F A365551 Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * f(s).
%F A365551 Sum_{k=1..n} a(k) ~ (Pi^2 * f(1) * n / 6) * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
%F A365551 f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...,
%F A365551 f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.452592479445159559037143959382854734148246511441192913672347667991...
%F A365551 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A365551 f[p_, e_] := Ceiling[(e + 3)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365551 (PARI) a(n) = vecprod(apply(x -> ceil((x+3)/2), factor(n)[, 2]));
%Y A365551 Cf. A322483, A356191.
%Y A365551 Cf. A049599, A282446, A353898.
%K A365551 nonn,easy,mult
%O A365551 1,2
%A A365551 _Amiram Eldar_, Sep 08 2023