A335032 - OEIS

%I #28 Dec 25 2022 02:11:38

%S 1,4,6,10,10,24,14,22,21,40,22,60,26,56,60,46,34,84,38,100,84,88,46,

%T 132,55,104,66,140,58,240,62,94,132,136,140,210,74,152,156,220,82,336,

%U 86,220,210,184,94,276,105,220,204,260,106,264,220,308,228,232,118

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

%C Dirichlet convolution of A000203 with abs(A097945).

%H Vaclav Kotesovec, <a href="/A335032/b335032.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A335032/a335032.jpg">Graph - the asymptotic ratio (1000000 terms)</a>.

%F Dirichlet g.f.: zeta(s) * zeta(s-1)^2 / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).

%F Dirichlet g.f.: zeta(s) * zeta(s-1)^2 * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).

%F 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.

%F a(n) = Sum_{d|n} A176345(d). - _Ridouane Oudra_, Jan 14 2022

%F Multiplicative with a(p^e) = sigma(p^e) + p^e - 1. - _Amiram Eldar_, Dec 25 2022

%t Table[Sum[DivisorSigma[1, n/d] * Abs[MoebiusMu[d]] * EulerPhi[d], {d, Divisors[n]}], {n, 1, 100}]

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)/(1 - p*X))[n], ", "))

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + p*X/(1 - X))/(1 - p*X))[n], ", "))

%Y Cf. A000203, A001620, A097945, A176345, A326306.

%K nonn,mult

%O 1,2

%A _Vaclav Kotesovec_, Jun 20 2020