OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).
FORMULA
Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Let f(s) = Product_{primes p} (1 - 1/(p^s + p)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)/2 + gamma - 3*zeta'(2)/Pi^2 - 1/4)*f(2) + f'(2)/2), where f(2) = A065463 = Product_{primes p} (1 - 1/(p*(p+1))) = 0.7044422009991655927366033503266372..., f'(2) = f(2) * Sum_{primes p} p*log(p) / ((p+1)*(p^2+p-1)) = 0.23219454323726621271960146689644280341444084188447499043209938838191022838..., for zeta'(2) see A073002 and gamma is the Euler-Mascheroni constant A001620.
a(n) = Sum_{d|n} A176345(d). - Ridouane Oudra, Jan 14 2022
Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - Amiram Eldar, Dec 25 2022
MATHEMATICA
Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Vaclav Kotesovec, Jun 20 2020
STATUS
approved