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A317358
a(n) is the smallest number k > 1 such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).
5
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)
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
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Jul 26 2018
EXTENSIONS
More terms from Michel Marcus, Jul 26 2018
STATUS
approved