A372996 a(n) = Sum_{k=1..n} sigma_2( (n/gcd(k,n))^2 ).

1, 22, 183, 704, 2605, 4026, 14707, 22548, 44469, 57310, 147631, 128832, 344773, 323554, 476715, 721596, 1340977, 978318, 2352295, 1833920, 2691381, 3247882, 6168163, 4126284, 8140625, 7585006, 10806147, 10353728, 19827445, 10487730, 27734491, 23091212, 27016473

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} phi(d) * sigma_2(d^2).

From Amiram Eldar, May 20 2024: (Start)

Multiplicative with a(p^e) = (p^(5*e+7) - p^(5*e+6) - p^(e+5) + p^e + p^6 - p^2) / ((p^2 - 1) * (p^5 - 1)).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(5) * zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 - 1/p^6 + 1/p^7) = 0.71416166953252012639... . (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A062380, A372227.

Cf. A013663, A013664.

Sequence in context: A197496 A107969 A248489 * A248490 A191012 A248491

Adjacent sequences: A372980 A372981 A372982 * A372997 A372998 A372999

Keyword

nonn,mult

Author

Seiichi Manyama, May 19 2024

Status

approved