A341637 a(n) = Sum_{d|n} phi(d) * sigma(d) * sigma(n/d).

1, 6, 12, 30, 30, 72, 56, 138, 123, 180, 132, 360, 182, 336, 360, 602, 306, 738, 380, 900, 672, 792, 552, 1656, 795, 1092, 1176, 1680, 870, 2160, 992, 2538, 1584, 1836, 1680, 3690, 1406, 2280, 2184, 4140, 1722, 4032, 1892, 3960, 3690, 3312, 2256, 7224, 2835, 4770, 3672, 5460

Offset: 1

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

• nonn
• mult

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[1, n/d], {d, Divisors[n]}], {n, 52}]
Table[Sum[DivisorSigma[1, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 52}]
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 *)

PARI

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*sigma(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

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

From Amiram Eldar, Nov 12 2022: (Start)

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

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)