OFFSET
1,3
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(2^e) = (4^e-1)/3, a(p^e) = (p^(2*e+1)+1)/(p+1), p>2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(4*zeta(2)) = 0.182690... (A240976). - Amiram Eldar, Oct 15 2022
Dirichlet g.f.: (zeta(s)*zeta(s-2)/zeta(s-1))*(1-2^(1-s)). - Amiram Eldar, Dec 30 2022
MATHEMATICA
f[p_, e_] := If[p == 2, (4^e - 1)/3, (p^(2*e + 1) + 1)/(p + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 15 2022 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, (4^f[i, 2]-1)/3, (f[i, 1]^(2*f[i, 2]+1)+1)/(f[i, 1]+1))); } \\ Amiram Eldar, Oct 15 2022
CROSSREFS
KEYWORD
mult,nonn
AUTHOR
Vladeta Jovovic, Dec 22 2002
STATUS
approved