A373000 a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^4 ).

1, 32, 243, 1054, 3125, 7776, 16807, 33818, 59289, 100000, 161051, 256122, 371293, 537824, 759375, 1082386, 1419857, 1897248, 2476099, 3293750, 4084101, 5153632, 6436343, 8217774, 9768745, 11881376, 14408187, 17714578, 20511149, 24300000, 28629151, 34636802, 39135393

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

If k is squarefree (cf. A005117) then a(k) = k^5.

a(n) = Sum_{d|n} phi(d) * sigma(d^4).

From Amiram Eldar, May 20 2024: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = (zeta(5) * zeta(6) / zeta(2)) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^5) = 1.00815225456201682259... . (End)

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^4] &]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^4));

Crossrefs

Cf. A005117.

Cf. A013663, A013664.

Sequence in context: A000584 A352051 A050752 * A351603 A343325 A343285

Adjacent sequences: A372997 A372998 A372999 * A373001 A373002 A373003

Keyword

nonn,mult

Author

Seiichi Manyama, May 19 2024

Status

approved