A365336 The sum of exponentially odd divisors of the square root of the largest square dividing n.
1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 11, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1
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^(2*Floor[(e+2)/4] + 1) - p)/(p^2 - 1) + 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]^(2*((f[i,2]+2)\4) + 1) - f[i,1])/(f[i,1]^2 - 1) + 1);}
Formula
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(2*floor((e+2)/4) + 1) - p)/(p^2 - 1) + 1. [corrected by Georg Fischer, Oct 07 2023]
Dirichlet g.f.: zeta(s) * zeta(4*s-2) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(4*s-2)).
From Vaclav Kotesovec, Sep 02 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(4*s-2) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(5*s-2)).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(4*s-2) * Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)).
Let f(s) = Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)), then Sum_{k=1..n} a(k) ~ Pi^2/12 * n * (f(1) * (log(n) + 3*gamma - 1 + 24*zeta'(2)/Pi^2) + f'(1)), where f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444021876..., f'(1) = f(1) * Sum_{primes p} 2*(4*p-3)*log(p) / (p^3 - 2*p + 1) = 1.617322217899181826790... and gamma is the Euler-Mascheroni constant A001620. (End)
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