A066808 a(n) = F(n)-1 mod 2^n+1 with F(n) = n-th Fermat number = 1+2^2^n.

0, 1, 1, 4, 1, 4, 16, 4, 1, 256, 16, 4, 4081, 4, 16, 256, 1, 4, 261121, 4, 65536, 256, 16, 4, 65536, 33554305, 16, 67108864, 65536, 4, 16, 4, 1, 256, 16, 262144, 68451041281, 4, 16, 256, 65536, 4, 4398042316801, 4, 65536, 35184371957761, 16, 4, 281474976645121

Offset: 0

Author

Wouter Meeussen, Jan 19 2002

Keywords

• nonn
• look

Comments

All terms except n=12,18,25,36,42,45,48,55 result in a(n) that are powers of 2, whereas these exceptions (4081, 261121, 33554305, 68451041281, 4398042316801, 35184371957761, 281474976645121, 36020000925941761) are all odd.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3323
• Chris Caldwell, Fermat Number, The Prime Glossary.
• Eric Weisstein's World of Mathematics, Fermat Number.

Crossrefs

Cf. A004249, A007516, A000215, A019434.

Programs

Maple

a:= n-> 2&^(2^n) mod (2^n+1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 04 2022

Mathematica

Table[ PowerMod[ 2, 2^n, 2^n+1 ], {n, 64} ]

Formula

F(n)-1=1 mod (2^n+1) for all n=2^k because F(n)=2+ F(1)F(2)..F(n-1)

Extensions

a(0)=0 prepended by Alois P. Heinz, Jul 04 2022