A341637 - OEIS

%I #14 Nov 12 2022 05:25:45

%S 1,6,12,30,30,72,56,138,123,180,132,360,182,336,360,602,306,738,380,

%T 900,672,792,552,1656,795,1092,1176,1680,870,2160,992,2538,1584,1836,

%U 1680,3690,1406,2280,2184,4140,1722,4032,1892,3960,3690,3312,2256,7224,2835,4770,3672,5460

%N a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

%H Michael De Vlieger, <a href="/A341637/b341637.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} sigma(gcd(n,k)) * sigma(n/gcd(n,k)).

%F From _Amiram Eldar_, Nov 12 2022: (Start)

%F Multiplicative with a(p^e) = (p^(2*e+3) - (e+1)*(p^2-1)*p^e - p)/((p-1)^2*(p+1)).

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/3) * A183699 * A330523 = 0.581007... . (End)

%t Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]

%t Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]

%t f[p_, e_] := (p^(2*e + 3) - (e + 1)*(p^2 - 1)*p^e - p)/((p - 1)^2*(p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 12 2022 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ _Michel Marcus_, Feb 17 2021

%Y Cf. A000010, A000203, A000385, A034761, A038040, A062354, A062952, A183699, A330523.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 16 2021