A351347 - OEIS

%I #20 Feb 11 2022 10:42:22

%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,

%T 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,

%U 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

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

%H Amiram Eldar, <a href="/A351347/b351347.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A351347/a351347.jpg">Graph - the asymptotic ratio (2*10^8 terms)</a>

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

%F From _Vaclav Kotesovec_, Feb 11 2022: (Start)

%F Let f(s) = Product_{prime p>2} (1 - 3/p^(2*s) + 2/p^(3*s))/(1 - 4/p^(2*s)), then

%F 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 f(1) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = A167864 = 1.5147801281374912577909192556494748924152701582862143953574842714849322098...,

%F 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)

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

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - X - 2*X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 10 2022

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

%K nonn,mult

%O 1,4

%A _Ilya Gutkovskiy_, Feb 08 2022