A317357 a(n) is the smallest composite k > n such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

4, 341, 473, 6, 10, 133, 497, 14, 12, 15, 15, 16, 18, 143, 35, 20, 32, 51, 57, 38, 28, 77, 253, 36, 30, 65, 39, 36, 58, 115, 155, 62, 36, 187, 119, 40, 74, 57, 247, 52, 80, 287, 2051, 86, 55, 69, 69, 94, 54, 175, 85, 65, 65, 159, 69, 70, 64, 551, 1711, 72

Offset: 1

Author

Thomas Ordowski, Jul 26 2018

Keywords

• nonn

Comments

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

a(n) > A151800(n) for all n < 33.

a(n) <= A271221(n) for n > 1.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..9661
• Wikipedia, Agoh-Giuga conjecture

Crossrefs

Cf. A151800, A271221, A317058, A317358.

Programs

Mathematica

a[n_] := Block[{k = n+1}, While[PrimeQ[k] || Mod[Sum[PowerMod[j, k-1, k], {j, n}], k] != n, k++]; k]; Array[a, 60] (* Giovanni Resta, Jul 26 2018 *)

PARI

a(n) = forcomposite(k=n+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 A317357(n):
    k = n+1
    while isprime(k) or g(n,k-1,k):
        k += 1
    return k # Chai Wah Wu, Jul 31 2018

Extensions

More terms from Giovanni Resta, Jul 26 2018