A372227 a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^2 ).

1, 8, 27, 70, 125, 216, 343, 578, 753, 1000, 1331, 1890, 2197, 2744, 3375, 4666, 4913, 6024, 6859, 8750, 9261, 10648, 12167, 15606, 15745, 17576, 20427, 24010, 24389, 27000, 29791, 37418, 35937, 39304, 42875, 52710, 50653, 54872, 59319, 72250, 68921, 74088

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

If k is squarefree (cf. A005117) then a(k) = k^3.

a(n) = Sum_{d|n} phi(d) * sigma(d^2).

From Amiram Eldar, May 20 2024: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = (Pi^2/15) * zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.03291869994469216597... . (End)

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^2] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^2));

Crossrefs

Cf. A062952, A372998, A373000.

Cf. A062380, A372996.

Cf. A005117.

Cf. A002117, A182448.

Sequence in context: A061096 A084825 A270806 * A246992 A351601 A276920

Adjacent sequences: A372224 A372225 A372226 * A372228 A372229 A372230

Keyword

nonn,mult

Author

Seiichi Manyama, May 19 2024

Status

approved