A365349 The sum of divisors of the smallest exponentially odd number divisible by n.
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
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)
Programs
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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]
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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
Formula
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)