A344138 a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n} gcd(x_1,x_2,x_3,x_4).

1, 17, 18, 99, 20, 306, 22, 373, 119, 340, 26, 1782, 28, 374, 360, 1115, 32, 2023, 34, 1980, 396, 442, 38, 6714, 165, 476, 532, 2178, 44, 6120, 46, 2901, 468, 544, 440, 11781, 52, 578, 504, 7460, 56, 6732, 58, 2574, 2380, 646, 62, 20070, 219, 2805, 576, 2772, 68, 9044, 520, 8206, 612, 748, 74, 35640, 76

Offset

1,2

Links

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

Formula

a(n) = Sum_{x_1|n, x_2|n, x_3|n, x_4|n} n/lcm(x_1,x_2,x_3,x_4).

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

If p is prime, a(p) = 15 + p.

a(n) = Sum_{k=1..n} tau(gcd(k,n))^4.

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * DivisorSigma[0, #]^4 &];  Array[a, 50] (* Amiram Eldar, May 10 2021 *)

Prog

(PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, gcd([i, j, k, l])))));
(PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, n/lcm([i, j, k, l])))));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d)^4);
(PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n))^4);

Crossrefs

Cf. A344132, A344139, A344140.

Sequence in context: A041600 A341113 A041598 * A041602 A041604 A041606

Adjacent sequences: A344135 A344136 A344137 * A344139 A344140 A344141

Keyword

nonn,mult

Author

Seiichi Manyama, May 10 2021

Status

approved