A351346 Dirichlet g.f.: Product_{p prime} 1 / (1 - 2*p^(-s) - p^(-2*s)).
1, 2, 2, 5, 2, 4, 2, 12, 5, 4, 2, 10, 2, 4, 4, 29, 2, 10, 2, 10, 4, 4, 2, 24, 5, 4, 12, 10, 2, 8, 2, 70, 4, 4, 4, 25, 2, 4, 4, 24, 2, 8, 2, 10, 10, 4, 2, 58, 5, 10, 4, 10, 2, 24, 4, 24, 4, 4, 2, 20, 2, 4, 10, 169, 4, 8, 2, 10, 4, 8, 2, 60, 2, 4, 10, 10, 4, 8, 2, 58, 29, 4, 2, 20, 4, 4, 4, 24, 2, 20
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)
Programs
-
Mathematica
f[p_, e_] := Fibonacci[e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 90}] -
PARI
for(n=1, 100, print1(direuler(p=2, n, 1/(1 - 2*X - X^2))[n], ", ")) \\ Vaclav Kotesovec, Feb 10 2022
Formula
Multiplicative with a(p^e) = Pell(e+1).
From Vaclav Kotesovec, Feb 11 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^s, where
s = log(1 + sqrt(2)) / log(2) = 1.271553303163611972...,
c = 8.3717222015175571... = (1 + sqrt(2)) / (2^(3/2) * log(1 + sqrt(2))) * Product_{p primes > 2} 1 / (1 - 2*p^(-s) - p^(-2*s)),
or with better convergence
c = zeta(s)^2 / (sqrt(2) * (1 + sqrt(2)) * log(1 + sqrt(2))) * Product_{p primes > 2} (1 - p^(-s))^2 / (1 - 2*p^(-s) - p^(-2*s)). (End)