A351347 Dirichlet g.f.: Product_{p prime} 1 / (1 - p^(-s) - 2*p^(-2*s)).

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

Offset: 1

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

Crossrefs

Cf. A001045, A061142, A351219, A351346, A351348.

Programs

Mathematica

f[p_, e_] := (2^(e + 1) + (-1)^e)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 100}]

PARI

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022

Formula

Multiplicative with a(p^e) = Jacobsthal(e+1).

From Vaclav Kotesovec, Feb 11 2022: (Start)

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)