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A365349 The sum of divisors of the smallest exponentially odd number divisible by n.

%I A365349 #17 Sep 04 2023 11:50:25
%S A365349 1,3,4,15,6,12,8,15,40,18,12,60,14,24,24,63,18,120,20,90,32,36,24,60,
%T A365349 156,42,40,120,30,72,32,63,48,54,48,600,38,60,56,90,42,96,44,180,240,
%U A365349 72,48,252,400,468,72,210,54,120,72,120,80,90,60,360,62,96,320
%N A365349 The sum of divisors of the smallest exponentially odd number divisible by n.
%H A365349 Amiram Eldar, <a href="/A365349/b365349.txt">Table of n, a(n) for n = 1..10000</a>
%H A365349 Vaclav Kotesovec, <a href="/A365349/a365349_1.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%F A365349 a(n) = A000203(A356191(n)).
%F A365349 Multiplicative with a(p^e) = (p^(e + 2 - (e mod 2)) - 1)/(p - 1).
%F A365349 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-3) - 1/p^(3*s-3)).
%F A365349 From _Vaclav Kotesovec_, Sep 04 2023: (Start)
%F A365349 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s-2) * zeta(2*s-3) * Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)).
%F A365349 Let f(s) = Product_{p prime} (1 - 1/p^(6*s-7) + 1/p^(5*s-6) + 1/p^(5*s-7) + 1/p^(4*s-4) + 1/p^(4*s-5) - 1/p^(4*s-6) - 1/p^(3*s-3) - 1/p^(3*s-4) - 1/p^(2*s-2)), then
%F A365349 Sum_{k=1..n} a(k) ~ n^2 * Pi^4 * f(2) / 144 * (log(n) + 3*gamma - 1/2 + 18*zeta'(2)/Pi^2 + f'(2)/f(2)), where
%F A365349 f(2) = Product_{p prime} (1 - 1/p^2) * (1 - 2/p^2 + 1/p^3) = 6*A065464/Pi^2 = 0.26034448085669554670553581687050222309091096557569931376863612821007515...,
%F A365349 f'(2) = f(2) * Sum_{p prime} 3*(3*p-2) * log(p) / (p^3 - 2*p + 1) = f(2) * 4.40861022247384449961018198035049309399000439627743168713608947117149645... and gamma is the Euler-Mascheroni constant A001620. (End)
%t A365349 f[p_, e_] := (p^(e + 2 - Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365349 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 2 - f[i,2]%2) - 1)/(f[i,1] - 1));}
%o A365349 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^3*X^3) )[n], ", ")) \\ _Vaclav Kotesovec_, Sep 04 2023
%Y A365349 Cf. A000203, A356191, A365348.
%K A365349 nonn,easy,mult
%O A365349 1,2
%A A365349 _Amiram Eldar_, Sep 02 2023