A342535 -id:A342535 - cp's OEIS frontend

A342539 a(n) = Sum_{k=1..n} phi(gcd(k, n))^n.

1, 2, 10, 19, 1028, 132, 279942, 65798, 10078726, 2097160, 100000000010, 16797702, 106993205379084, 156728328204, 35186519703560, 281479271809036, 295147905179352825872, 203119914385420, 708235345355337676357650, 1152924803145924620, 46005163783270994804748, 20000000000000000000020

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..388

Cf. A000010, A029935, A332517, A342487, A342534, A342535, A342540, A342541, A342543.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)

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

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

a(n) = Sum_{d|n} phi(n/d) * phi(d)^n.
If p is prime, a(p) = p-1 + (p-1)^p.
a(n) = Sum_{k=1..n} phi(n/gcd(n,k))^(n-1)*phi(gcd(n,k)). - Richard L. Ollerton, May 09 2021

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

Original entry on oeis.org

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

Seiichi Manyama, Mar 15 2021

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

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

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

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

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

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)

cp's OEIS Frontend

A342539 a(n) = Sum_{k=1..n} phi(gcd(k, n))^n.

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

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