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A365349
The sum of divisors of the smallest exponentially odd number divisible by n.
4
1, 3, 4, 15, 6, 12, 8, 15, 40, 18, 12, 60, 14, 24, 24, 63, 18, 120, 20, 90, 32, 36, 24, 60, 156, 42, 40, 120, 30, 72, 32, 63, 48, 54, 48, 600, 38, 60, 56, 90, 42, 96, 44, 180, 240, 72, 48, 252, 400, 468, 72, 210, 54, 120, 72, 120, 80, 90, 60, 360, 62, 96, 320
OFFSET
1,2
FORMULA
a(n) = A000203(A356191(n)).
Multiplicative with a(p^e) = (p^(e + 2 - (e mod 2)) - 1)/(p - 1).
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)).
From Vaclav Kotesovec, Sep 04 2023: (Start)
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)).
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
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(2) = Product_{p prime} (1 - 1/p^2) * (1 - 2/p^2 + 1/p^3) = 6*A065464/Pi^2 = 0.26034448085669554670553581687050222309091096557569931376863612821007515...,
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)
MATHEMATICA
f[p_, e_] := (p^(e + 2 - Mod[e, 2]) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(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)); }
(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
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 02 2023
STATUS
approved