A351348 Dirichlet g.f.: Product_{p prime} (1 + 2*p^(-s)) / (1 - p^(-s) - p^(-2*s)).

1, 3, 3, 4, 3, 9, 3, 7, 4, 9, 3, 12, 3, 9, 9, 11, 3, 12, 3, 12, 9, 9, 3, 21, 4, 9, 7, 12, 3, 27, 3, 18, 9, 9, 9, 16, 3, 9, 9, 21, 3, 27, 3, 12, 12, 9, 3, 33, 4, 12, 9, 12, 3, 21, 9, 21, 9, 9, 3, 36, 3, 9, 12, 29, 9, 27, 3, 12, 9, 27, 3, 28, 3, 9, 12, 12, 9, 27, 3, 33, 11, 9, 3, 36, 9, 9, 9

Offset

1,2

Links

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

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

Formula

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

a(n) = Sum_{d|n} A074823(d) * A351219(n/d).

From Vaclav Kotesovec, Feb 12 2022: (Start)

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

Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + ((3*g-1)*f(1) + f'(1))*log(n) + (1 - 3*g + 3*g^2 - 3*sg1)*f(1) + (3*g-1)*f'(1) + f''(1)/2), where

f(1) = Product_{prime p} (p-1)^3 * (p+2) / (p^2 (p^2 - p - 1)) = 0.76679494740111861346654669603448358442373234633770198438779408968851774...,

f'(1) = f(1) * Sum_{p prime} (4*p^2 - 9*p - 4) * log(p) / (p^4 - 4*p^2 + p + 2) = -0.2518173642312369311596467494348076414732211832249275289370643712012051...,

f''(1) = f'(1)^2/f(1) + f(1) * Sum_{p prime} -p*(8*p^5 - 27*p^4 - 16*p^3 + 32*p^2 + 16*p + 14) * log(p)^2 / (p^4 - 4*p^2 + p + 2)^2 = 4.28643633804365513728313780779157573071314496047204449783182235740130206...,

gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

Mathematica

f[p_, e_] := LucasL[e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 87}]

Prog

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

Crossrefs

Cf. A000204, A074823, A351219, A351346, A351347.

Sequence in context: A326401 A326415 A343443 * A246011 A329310 A061023

Adjacent sequences: A351345 A351346 A351347 * A351349 A351350 A351351

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 08 2022

Status

approved