A342470 -id:A342470 - cp's OEIS frontend

A342471 a(n) = Sum_{d|n} phi(d)^n.

1, 2, 9, 18, 1025, 130, 279937, 65794, 10078209, 2097154, 100000000001, 16789506, 106993205379073, 156728328194, 35185445863425, 281479271743490, 295147905179352825857, 203119913861122, 708235345355337676357633, 1152923703631151106

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

Cf. A000010, A029939, A110567, A309369, A338997, A342470.

a[n_] := DivisorSum[n, EulerPhi[#]^n &]; Array[a, 20] (* Amiram Eldar, Mar 13 2021 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(d)^n);
```

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-(eulerphi(k)*x)^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n-1).
G.f.: Sum_{k>=1} (phi(k)*x)^k/(1 - (phi(k)*x)^k).
If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^n/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A342473 a(n) = Sum_{d|n} phi(d)^d.

Original entry on oeis.org

1, 2, 9, 18, 1025, 74, 279937, 65554, 10077705, 1049602, 100000000001, 16777306, 106993205379073, 78364444034, 35184372089865, 281474976776210, 295147905179352825857, 101559966746186, 708235345355337676357633, 1152921504607896594, 46005119909369701746057, 10000000000100000000002

Offset: 1

Seiichi Manyama, Mar 13 2021

nonn

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

Cf. A000010, A029939, A110567, A309369, A338997, A342470, A342471, A342485.

a[n_] := DivisorSum[n, EulerPhi[#]^# &]; Array[a, 20] (* Amiram Eldar, Mar 14 2021 *)

PARI
```
a(n) = sumdiv(n, d, eulerphi(d)^d);
```

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

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (eulerphi(k)*x)^k/(1-x^k)))

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(n/gcd(k, n) - 1).
G.f.: Sum_{k>=1} (phi(k) * x)^k/(1 - x^k).
If p is prime, a(p) = 1 + (p-1)^p = A110567(p-1).
a(n) = Sum_{k=1..n} phi(gcd(n,k))^gcd(n,k)/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

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

Original entry on oeis.org

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

Seiichi Manyama, Mar 15 2021

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

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

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 *)

a(n) = sum(k=1, n, eulerphi(gcd(k, n))^3);

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^3);

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

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A342471 a(n) = Sum_{d|n} phi(d)^n.

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A342473 a(n) = Sum_{d|n} phi(d)^d.

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

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