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

0,4

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.

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)

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} ]

Crossrefs

Cf. A004249, A007516, A000215, A019434.

Sequence in context: A205125 A248978 A143461 * A363935 A033918 A136467

Adjacent sequences: A066805 A066806 A066807 * A066809 A066810 A066811

Keyword

nonn,look

Author

Wouter Meeussen, Jan 19 2002

Extensions

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

Status

approved