A078747 Expansion of Sum_{k>0} k*phi(k)*x^k/(1+x^k).

1, 1, 7, 5, 21, 7, 43, 21, 61, 21, 111, 35, 157, 43, 147, 85, 273, 61, 343, 105, 301, 111, 507, 147, 521, 157, 547, 215, 813, 147, 931, 341, 777, 273, 903, 305, 1333, 343, 1099, 441, 1641, 301, 1807, 555, 1281, 507, 2163, 595, 2101, 521, 1911, 785, 2757, 547

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

Cf. A000010, A057660, A240976, A299069.

Sequence in context: A204138 A179118 A166639 * A213835 A145396 A263825

Adjacent sequences: A078744 A078745 A078746 * A078748 A078749 A078750

Keyword

mult,nonn

Author

Vladeta Jovovic, Dec 22 2002

Status

approved