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A365348 The number of divisors of the smallest exponentially odd number divisible by n.

%I A365348 #10 Sep 05 2023 05:34:47
%S A365348 1,2,2,4,2,4,2,4,4,4,2,8,2,4,4,6,2,8,2,8,4,4,2,8,4,4,4,8,2,8,2,6,4,4,
%T A365348 4,16,2,4,4,8,2,8,2,8,8,4,2,12,4,8,4,8,2,8,4,8,4,4,2,16,2,4,8,8,4,8,2,
%U A365348 8,4,8,2,16,2,4,8,8,4,8,2,12,6,4,2,16,4
%N A365348 The number of divisors of the smallest exponentially odd number divisible by n.
%C A365348 The sum of these divisors is A365349(n).
%H A365348 Amiram Eldar, <a href="/A365348/b365348.txt">Table of n, a(n) for n = 1..10000</a>
%H A365348 Vaclav Kotesovec, <a href="/A365348/a365348.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%F A365348 a(n) = A000005(A356191(n)).
%F A365348 Multiplicative with a(p^e) = e + 2 - (e mod 2).
%F A365348 Dirichlet g.f.: zeta(s)^2 * zeta(2*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
%F A365348 From _Vaclav Kotesovec_, Sep 05 2023: (Start)
%F A365348 Let f(s) = Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s)).
%F A365348 Sum_{k=1..n} a(k) ~ f(1) * Pi^2 * n / 6 * (log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 + f'(1)/f(1)), where
%F A365348 f(1) = Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.74469549790606742043912387159445432811796913290492411186307181370150975026...
%F A365348 f'(1) = f(1) * Sum_{p prime} 2*(3*p - 2) * log(p) / (1 - 2*p + p^4) = f(1) * 0.75575434641494973924789411019492794958528241212857430737760075121773728338...
%F A365348 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A365348 f[p_, e_] := e + 2 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365348 (PARI) a(n) = vecprod(apply(x -> x + 2 - x%2, factor(n)[, 2]));
%Y A365348 Cf. A000005, A356191, A365349.
%K A365348 nonn,easy,mult
%O A365348 1,2
%A A365348 _Amiram Eldar_, Sep 02 2023