A231565 -id:A231565 - cp's OEIS frontend

A231563 a(n) = f(1)^n + ... + f(n)^n (mod n) where f(i)=i if gcd(i,n)=1 and f(i)=0 otherwise.

0, 1, 0, 2, 0, 2, 0, 4, 0, 0, 0, 4, 0, 0, 0, 8, 0, 6, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 20, 0, 16, 0, 0, 0, 12, 0, 0, 0, 16, 0, 12, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 16, 0, 0, 0, 32, 0, 44, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0

Offset: 1

José María Grau Ribas, Dec 09 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A231564, A231565, A308481.

S[n_] :=  Mod[Sum[If[GCD[i, n] == 1, PowerMod[i, n, n], 0], {i, 1, n}], n]; Array[S,100]

f(i, n) = if (gcd(i, n) == 1, i, 0);
a(n) = lift(sum(k=1, n, Mod(f(k, n), n)^n)); \\ Michel Marcus, Jul 16 2017

a(n) = A308481(n) mod n. - Seiichi Manyama, Feb 11 2021

A231564 Numbers such that A231563(n)=0.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88

Offset: 1

José María Grau Ribas, Dec 09 2013

nonn

This sequence contains the odd primes.

Cf. A231563, A231565.

S[n_] := Mod[Sum[If[GCD[i, n] == 1, PowerMod[i, n, n], 0], {i, 1, n}], n];Select[Range[100],S[#]==0&]

A263014 a(n) = Sum_{0 < a, b <= n and gcd(a^2 + b^2, n) = 1} (a + bi)^n (mod n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0

Offset: 1

José María Grau Ribas, Oct 07 2015

nonn

Sum of the n-th powers of the invertible elements of Z[i]/nZ[i].

Hans Havermann, Table of n, a(n) for n = 1..5000

See A263016 for indices where this is nonzero.
See A290287 for the nonzero values.
Cf. A231563, A231564, A231565.

Sp[n_, k_] := Mod[Sum[If[GCD[a^2 + b^2, n] == 1, PowerMod[(a + b I), k, n], 0], {a, n}, {b, n}], n]; Table[ Sp[n, n] , {n, 1, 74}]

cp's OEIS Frontend

A231563 a(n) = f(1)^n + ... + f(n)^n (mod n) where f(i)=i if gcd(i,n)=1 and f(i)=0 otherwise.

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A231564 Numbers such that A231563(n)=0.

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A263014 a(n) = Sum_{0 < a, b <= n and gcd(a^2 + b^2, n) = 1} (a + bi)^n (mod n).

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