A342544 -id:A342544 - cp's OEIS frontend

A342540 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n-1).

1, 2, 6, 11, 260, 68, 46662, 16518, 1680134, 524296, 10000000010, 4204550, 8916100448268, 26121388044, 4398583447560, 35185445896204, 18446744073709551632, 33853319413772, 39346408075296537575442, 144116012711673868, 3833767304764361539596, 2000000000000000000020

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

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

Cf. A000010, A029935, A342471, A342539, A342542, A342544.

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

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

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

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

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

A342542 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 9, 15, 13, 11, 33, 13, 19, 105, 33, 17, 91, 19, 209, 469, 31, 23, 641, 1045, 37, 1627, 841, 29, 4217, 31, 673, 10461, 49, 29785, 10281, 37, 55, 49465, 68769, 41, 65197, 43, 12281, 529625, 67, 47, 273185, 279979, 1049661, 1049121, 52657, 53, 803647

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

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

Cf. A000010, A029935, A342473, A342540, A342541, A342544.

a[n_] := DivisorSum[n, EulerPhi[n/#] * EulerPhi[#]^(n/#-1) &]; Array[a, 50] (* Amiram Eldar, Mar 15 2021 *)

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

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

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

If p is prime, a(p) = p.

A342543 a(n) = Sum_{k=1..n} phi(gcd(k, n))^gcd(k, n).

Original entry on oeis.org

1, 2, 10, 19, 1028, 76, 279942, 65558, 10077718, 1049608, 100000000010, 16777334, 106993205379084, 78364444044, 35184372090920, 281474976776236, 295147905179352825872, 101559966746268, 708235345355337676357650, 1152921504607897676, 46005119909369702026044, 10000000000100000000020

Offset: 1

Seiichi Manyama, Mar 15 2021

nonn

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

Cf. A000010, A029935, A342423, A342488, A342539, A342541, A342544.

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

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

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

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

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

cp's OEIS Frontend

A342540 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n-1).

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A342542 a(n) = Sum_{k=1..n} phi(gcd(k, n))^(n/gcd(k, n) - 1).

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Mathematica

PARI

PARI

Formula

A342543 a(n) = Sum_{k=1..n} phi(gcd(k, n))^gcd(k, n).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula