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A066808
a(n) = F(n)-1 mod 2^n+1 with F(n) = n-th Fermat number = 1+2^2^n.
1
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
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
KEYWORD
nonn,look
AUTHOR
Wouter Meeussen, Jan 19 2002
EXTENSIONS
a(0)=0 prepended by Alois P. Heinz, Jul 04 2022
STATUS
approved