A342543 -id:A342543 - 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

A342541 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n)).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 12, 14, 28, 28, 20, 62, 24, 54, 272, 68, 32, 198, 36, 676, 1224, 130, 44, 1348, 4136, 180, 3540, 3426, 56, 12632, 60, 1640, 22520, 304, 129456, 22370, 72, 378, 101808, 270952, 80, 192996, 84, 40630, 1867184, 550, 92, 551528, 1679700, 4198860, 2105408

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A000010, A029935, A342485, A342539, A342542, A342543.

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

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

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

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

If p is prime, a(p) = 2 *(p-1).

A342544 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(gcd(k, n) - 1).

Original entry on oeis.org

1, 2, 6, 11, 260, 40, 46662, 16398, 1679630, 262408, 10000000010, 4194366, 8916100448268, 13060740684, 4398046511640, 35184372105244, 18446744073709551632, 16926661124436, 39346408075296537575442, 144115188076118572, 3833759992447475215524, 1000000000010000000020

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

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

Cf. A000010, A029935, A309369, A342436, A342540, A342542, A342543.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(# - 1) &]; Array[a, 20] (* Amiram Eldar, Mar 15 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*eulerphi(d)^(d-1));

a(n) = Sum_{d|n} phi(n/d) * phi(d)^(d-1).

If p is prime, a(p) = p-1 + (p-1)^(p-1).

cp's OEIS Frontend

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

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A342541 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n)).

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Formula

A342544 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(gcd(k, n) - 1).

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Mathematica

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