A342534 a(n) = Sum_{k=1..n} phi(gcd(k, n))^2.

1, 2, 6, 7, 20, 12, 42, 26, 50, 40, 110, 42, 156, 84, 120, 100, 272, 100, 342, 140, 252, 220, 506, 156, 484, 312, 438, 294, 812, 240, 930, 392, 660, 544, 840, 350, 1332, 684, 936, 520, 1640, 504, 1806, 770, 1000, 1012, 2162, 600, 2022, 968, 1632, 1092, 2756, 876, 2200, 1092, 2052

Offset: 1

Author

Seiichi Manyama, Mar 15 2021

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000010, A029935, A029939, A065464, A306633, A338997, A342535.

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^2 &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

PARI

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^2);

PARI

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

Formula

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

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

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = (p-1)*(p^(e-2) - p^(2*e-3) + p^(2*e-1)).

Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)/(3*zeta(3)) * Product_{p prime} (1 - (2*p-1)/p^3) = A306633 * A065464 / 3 = 0.195343... . (End)