OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
FORMULA
Let f(s) = Product_{primes p} (p^s-1)^2 * (p^(2*s)-p) * (p^(2*s) + 2*p^s + p) / p^(6*s).
Dirichlet g.f.: zeta(s)^2 * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n)^2 + 2*(4*gamma - 1 + f'(1)/f(1))*log(n) + 2 - 8*gamma + 10*gamma^2 - 12*sg1 + (2*(4*gamma - 1)*f'(1) + f''(1))/f(1)) / 4, where
f(1) = Product_{primes p} (1 - 6/p^2 + 8/p^3 - 3/p^4) = A319592 = 0.1148840440802287887292512767015990978487135526872830176248484...,
f'(1) = f(1) * Sum_{primes p} 16*log(p)/((p-1)*(p+3)) = 0.7329537986688178103828762539130015065766096293897775989858128...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} -2*(23*p^2 + 26*p - 1) * log(p)^2 / ((p-1)^2 * (p+3)^2) = 1.3073198533301549336255068891648606199737582186092200289304...,
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633).
Multiplicative with a(p) = 2, a(p^2) = p, and a(p^e) = 0 for e >= 3. - Amiram Eldar, Jul 21 2026
MATHEMATICA
f[p_, 1] := 2; f[p_, 2] := p; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 21 2026 *)
PROG
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X + p*X^2))[n], ", "))
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Vaclav Kotesovec, Jul 20 2026
STATUS
approved