A033457 - OEIS

%I #24 Dec 06 2024 07:35:46

%S 1,2,6,4,19,6,28,24,45,10,98,12,79,94,120,16,201,18,238,164,171,22,

%T 436,120,229,234,426,28,695,30,496,352,369,370,1014,36,451,470,1068,

%U 40,1261,42,946,1020,639,46,1832,336,1225,754,1278,52,1899,774,1924,920,981

%N GCD-convolution of squares A000290 with themselves.

%H Danny Rorabaugh, <a href="/A033457/b033457.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n-2) = Sum_{d|n, d<n} d^2*phi(n/d). - _Vladeta Jovovic_, Aug 27 200

%F From _Amiram Eldar_, Dec 06 2024: (Start)

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

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

%t Table[Sum[d^2*EulerPhi[(n + 2)/d], {d, Most@ Divisors[n + 2]}], {n, 0, 47}] (* _Michael De Vlieger_, Mar 20 2015 *)

%t 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 *)

%o (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

%o (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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_