A342535 a(n) = Sum_{k=1..n} phi(gcd(k, n))^3.

1, 2, 10, 11, 68, 20, 222, 78, 238, 136, 1010, 110, 1740, 444, 680, 604, 4112, 476, 5850, 748, 2220, 2020, 10670, 780, 8276, 3480, 6330, 2442, 21980, 1360, 27030, 4792, 10100, 8224, 15096, 2618, 46692, 11700, 17400, 5304, 64040, 4440, 74130, 11110, 16184, 21340, 97382, 6040

Offset

1,2

Links

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

Formula

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

a(n) = Sum_{k=1..n} phi(gcd(k,n))*phi(n/gcd(k,n))^2. - Richard L. Ollerton, May 10 2021

Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/4) * Product_{p prime} (1 - 3/p^2 + 3/p^3 - 2/p^4 + 3/p^6 - 3/p^7 + 1/p^8) = 0.093622450005... . - Amiram Eldar, Nov 15 2022

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^3 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)
Table[Sum[EulerPhi[GCD[k, n]]^3, {k, n}], {n, 50}] (* Harvey P. Dale, Jul 15 2021 *)

Prog

(PARI) a(n) = sum(k=1, n, eulerphi(gcd(k, n))^3);
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^3);

Crossrefs

Cf. A000010, A002117, A029935, A342470, A342534.

Sequence in context: A174570 A072545 A023151 * A265747 A104459 A008560

Adjacent sequences: A342532 A342533 A342534 * A342536 A342537 A342538

Keyword

nonn,mult

Author

Seiichi Manyama, Mar 15 2021

Status

approved