OFFSET
0,4
COMMENTS
Conjectures:
1. Indices of zeros: 2^(x+2)-2, x >= 0.
2. a(n)=n if n is in A176003.
3. Every integer k >= 0 appears in a(n) at least once.
4. Every k >= 0 appears in a(n) infinitely many times.
From Robert Israel, May 05 2015: (Start)
Conjecture 1) is true: with m = n+2, a(n) = (-2)^(m-2) mod m = 0 iff m divides 2^(m-2), i.e., m = 2^k for some k with k <= m-2 (which is true for k >= 2).
Conjecture 2) is true: if n = 3*p-2 where p is prime, then n == 1 (mod 3) so n^n == n (mod 3), and n^(p-1) == 1 (mod p) so n^n == n (mod p), and therefore (if p <> 3) n^n == n (mod 3*p). A separate computation verifies the case p=3.
If p is an odd prime, then a(p+2) = (p-1)/2. (End)
LINKS
Paolo P. Lava, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (n^n) mod (n+2).
a(n) = (-2)^n mod (n+2). - Robert Israel, May 05 2015
EXAMPLE
a(5) = 5^5 mod 7 = 3125 mod 7 = 3.
MATHEMATICA
a[n_] := PowerMod[-2, n, n+2];
a /@ Range[0, 100] (* Jean-François Alcover, Jun 04 2020 *)
Table[PowerMod[n, n, n+2], {n, 0, 80}] (* Harvey P. Dale, Oct 23 2024 *)
PROG
(PARI) a(n) = lift(Mod(n, n+2)^n); \\ Michel Marcus, Jun 04 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Jun 10 2012
STATUS
approved