A317058 a(n) is the smallest composite k such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

4, 341, 473, 4, 4, 133, 497, 4, 4, 15, 9, 4, 4, 143, 35, 4, 4, 51, 57, 4, 4, 77, 253, 4, 4, 65, 9, 4, 4, 115, 155, 4, 4, 187, 35, 4, 4, 9, 247, 4, 4, 287, 2051, 4, 4, 15, 33, 4, 4, 35, 85, 4, 4, 9, 9, 4, 4, 551, 1711, 4, 4, 713, 21, 4, 4, 55, 77, 4, 4, 35, 35, 4

Offset

1,1

Comments

According to the Agoh-Giuga conjecture, a(n) <> n+1.

a(n) = 4 if and only if n == {0, 1} (mod 4).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..9661 (n = 1..500 from Seiichi Manyama)

Wikipedia, Agoh-Giuga conjecture

Mathematica

a[n_] := Block[{k = 4}, While[PrimeQ[k] || Mod[Sum[PowerMod[j, k-1, k], {j, n}], k] != Mod[n, k], k++]; k]; Array[a, 72] (* Giovanni Resta, Jul 26 2018 *)

Prog

(PARI) a(n) = forcomposite(k=1, , if (sum(j=1, n, Mod(j, k)^(k-1)) == n, return (k)); ); \\ Michel Marcus, Jul 26 2018
(Python)
from sympy import isprime
def g(n, p, q): # compute (-n + sum_{k=1, n} k^p)  mod q
    c = (-n) % q
    for k in range(1, n+1):
        c = (c+pow(k, p, q)) % q
    return c
def A317058(n):
    k = 2
    while isprime(k) or g(n, k-1, k):
        k += 1
    return k # Chai Wah Wu, Jul 30 2018

Crossrefs

Cf. A000790, A133906, A133907, A317357, A317358.

Sequence in context: A007535 A000783 A098654 * A317357 A069884 A332134

Adjacent sequences: A317055 A317056 A317057 * A317059 A317060 A317061

Keyword

nonn

Author

Thomas Ordowski, Jul 26 2018

Extensions

More terms from Giovanni Resta, Jul 26 2018

Status

approved