A317358 a(n) is the smallest number k > 1 such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

2, 3, 5, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 29, 2, 2, 5, 3, 2, 2, 31, 37, 2, 2, 37, 35, 2, 2, 3, 41, 2, 2, 43, 47, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 59, 61, 2, 2, 67, 3, 2, 2, 55, 71, 2, 2, 35, 35, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2

Offset

1,1

Comments

a(n) = 2 if and only if n == {0, 1} (mod 4).

a(n) <= A151800(n).

A133906(n) <= a(n) <= A133907(n).

The sequence is unbounded.

Numbers n such that a(n-1) = n are 2, 3, 7, 23, 31, 43, 59, 139, 283, ...

By the Agoh-Giuga conjecture, if a(n-1) = n, then n is a prime.

It seems that if a(n) > n, then a(n) is a prime (the next prime after n).

If a(n) = n, then n is in A121707. These numbers are 35, 143, 187, 215, ...

Conjecture: all composite terms of the sequence are A121707.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Seiichi Manyama)

Wikipedia, Agoh-Giuga conjecture

Mathematica

a[n_] := Block[{k=2}, While[Mod[Sum[PowerMod[j, k-1, k], {j, n}], k] != Mod[n, k], k++]; k]; Array[a, 81] (* Giovanni Resta, Jul 29 2018 *)

Prog

(PARI) a(n) = for(k=2, oo, if (sum(j=1, n, Mod(j, k)^(k-1)) == n, return (k)); ); \\ Michel Marcus, Jul 26 2018
(Python)
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 A317358(n):
    k = 2
    while g(n, k-1, k):
        k += 1
    return k # Chai Wah Wu, Jul 30 2018

Crossrefs

Cf. A133906, A133907, A151800, A121707, A317058, A317357.

Sequence in context: A255913 A065996 A133906 * A133907 A232931 A060084

Adjacent sequences: A317355 A317356 A317357 * A317359 A317360 A317361

Keyword

nonn

Author

Thomas Ordowski, Jul 26 2018

Extensions

More terms from Michel Marcus, Jul 26 2018

Status

approved