A033457 GCD-convolution of squares A000290 with themselves.

1, 2, 6, 4, 19, 6, 28, 24, 45, 10, 98, 12, 79, 94, 120, 16, 201, 18, 238, 164, 171, 22, 436, 120, 229, 234, 426, 28, 695, 30, 496, 352, 369, 370, 1014, 36, 451, 470, 1068, 40, 1261, 42, 946, 1020, 639, 46, 1832, 336, 1225, 754, 1278, 52, 1899, 774, 1924, 920, 981

Offset

0,2

Links

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Formula

a(n-2) = Sum_{d|n, d<n} d^2*phi(n/d). - Vladeta Jovovic, Aug 27 200

From Amiram Eldar, Dec 06 2024: (Start)

a(n) = A069097(n+2) - (n+2)^2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)/zeta(3) - 1)/3 = (A306633 - 1)/3 = 0.122810925... . (End)

Mathematica

Table[Sum[d^2*EulerPhi[(n + 2)/d], {d, Most@ Divisors[n + 2]}], {n, 0, 47}] (* Michael De Vlieger, Mar 20 2015 *)
f[p_, e_] := p^(e - 1)*(p^e*(p + 1) - 1); a[n_] := Times @@ f @@@ FactorInteger[n + 2] - (n + 2)^2; Array[a, 100, 0] (* Amiram Eldar, Dec 06 2024 *)

Prog

(Sage) sum([d^2*euler_phi(int((n+2)/d)) for d in range(1, n+2) if (n+2)%d==0]) # Danny Rorabaugh, Mar 20 2015
(PARI) a(n) = {my(f = factor(n+2)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; p^(e-1)*(p^e*(p+1) - 1)) - (n+2)^2; } \\ Amiram Eldar, Dec 06 2024

Crossrefs

Cf. A000010 (phi), A000290, A069097, A306633.

Sequence in context: A054786 A269372 A282902 * A133936 A065350 A333923

Adjacent sequences: A033454 A033455 A033456 * A033458 A033459 A033460

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved