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

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Vaclav Kotesovec, Graph - the asymptotic ratio (2*10^8 terms)

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)

Mathematica

f[p_, e_] := Fibonacci[e + 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 90}]

Prog

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

Crossrefs

Cf. A000129, A351219, A351347, A351348.

Sequence in context: A339664 A124316 A328705 * A061034 A245635 A111861

Adjacent sequences: A351343 A351344 A351345 * A351347 A351348 A351349

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 08 2022

Status

approved