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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Programs

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 *)

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

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

Formula

a(n-2) = Sum_{d|n, dVladeta Jovovic, Aug 27 2003

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)