A344132 a(n) = Sum_{i|n, j|n, k|n} gcd(i,j,k).

1, 9, 10, 37, 12, 90, 14, 111, 49, 108, 18, 370, 20, 126, 120, 283, 24, 441, 26, 444, 140, 162, 30, 1110, 79, 180, 184, 518, 36, 1080, 38, 657, 180, 216, 168, 1813, 44, 234, 200, 1332, 48, 1260, 50, 666, 588, 270, 54, 2830, 117, 711, 240, 740, 60, 1656, 216, 1554, 260, 324, 66, 4440, 68, 342, 686, 1441, 240, 1620, 74

Offset

1,2

Links

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

Formula

a(n) = Sum_{i|n, j|n, k|n} n/lcm(i,j,k).

a(n) = Sum_{d|n} phi(n/d) * tau(d)^3.

If p is prime, a(p) = 7 + p.

a(n) = Sum_{k=1..n} tau(gcd(k,n))^3.

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^3 &]; Array[a, 50] (* Amiram Eldar, May 10 2021 *)

Prog

(PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, gcd([i, j, k]))));
(PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, n/lcm([i, j, k]))));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^3);
(PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^3);

Crossrefs

Cf. A060724, A344133, A344134, A344135, A344138, A344139, A344140.

Sequence in context: A329636 A289520 A038206 * A154389 A041172 A248353

Adjacent sequences: A344129 A344130 A344131 * A344133 A344134 A344135

Keyword

nonn,mult

Author

Seiichi Manyama, May 10 2021

Status

approved