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A351347
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Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2*p^(-2*s)).
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6
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1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 3, 1, 1, 1, 11, 1, 3, 1, 3, 1, 1, 1, 5, 3, 1, 5, 3, 1, 1, 1, 21, 1, 1, 1, 9, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 11, 3, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 43, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 3, 3, 1, 1, 1, 11, 11, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 21, 1, 3, 3, 9
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OFFSET
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1,4
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LINKS
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FORMULA
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Multiplicative with a(p^e) = Jacobsthal(e+1).
Let f(s) = Product_{prime p>2} (1 - 3/p^(2*s) + 2/p^(3*s))/(1 - 4/p^(2*s)), then
Sum_{k=1..n} a(k) ~ n*((2 * Pi^2 * log(n) + Pi^2 * (5*log(2) + 2*gamma - 2) + 24*zeta'(2))*f(1) + 2*Pi^2 * f'(1)) / (48*log(2)), where
f(1) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = A167864 = 1.5147801281374912577909192556494748924152701582862143953574842714849322098...,
f'(1) = -f(1) * Sum_{primes p > 2} 2*log(p) / (2 - 3*p + p^2) = -2*f(1)*A347195 = -2.603580548675394425068281893203286277011306183054394825715911358402698051... and gamma is the Euler-Mascheroni constant A001620. (End)
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MATHEMATICA
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f[p_, e_] := (2^(e + 1) + (-1)^e)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}]
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PROG
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(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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