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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.

%I A066808 #18 Feb 16 2025 08:32:45
%S A066808 0,1,1,4,1,4,16,4,1,256,16,4,4081,4,16,256,1,4,261121,4,65536,256,16,
%T A066808 4,65536,33554305,16,67108864,65536,4,16,4,1,256,16,262144,
%U A066808 68451041281,4,16,256,65536,4,4398042316801,4,65536,35184371957761,16,4,281474976645121
%N A066808 a(n) = F(n)-1 mod 2^n+1 with F(n) = n-th Fermat number = 1+2^2^n.
%C A066808 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.
%H A066808 Alois P. Heinz, <a href="/A066808/b066808.txt">Table of n, a(n) for n = 0..3323</a>
%H A066808 Chris Caldwell, <a href="https://primes.utm.edu/glossary/page.php?sort=FermatNumber">Fermat Number</a>, The Prime Glossary.
%H A066808 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%F A066808 F(n)-1=1 mod (2^n+1) for all n=2^k because F(n)=2+ F(1)F(2)..F(n-1)
%p A066808 a:= n-> 2&^(2^n) mod (2^n+1):
%p A066808 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 04 2022
%t A066808 Table[ PowerMod[ 2, 2^n, 2^n+1 ], {n, 64} ]
%Y A066808 Cf. A004249, A007516, A000215, A019434.
%K A066808 nonn,look
%O A066808 0,4
%A A066808 _Wouter Meeussen_, Jan 19 2002
%E A066808 a(0)=0 prepended by _Alois P. Heinz_, Jul 04 2022