A066808 - OEIS

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

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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