OFFSET
1,1
COMMENTS
The corresponding values of the Fermat pseudoprime p: 1729, 2047, 3277, 8911, 10585, 13747, 14491, 15709, 16705, 41041, 49141, 60787, 88357, 196093, 215749, 223345, 256999, 278545, 401401, 449065, 657901.
Conjecture: For any Fermat pseudoprime to base 2, p1, there exist infinitely many Fermat pseudoprimes to base 2, of the form p2 = (p1^n + n*p1)/(n+1), where n > 1.
Conjecture: For any Carmichael number c there exist infinitely many Carmichael numbers of the form (c^n + n*c)/(n + 1) with n > 1.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Poulet Number
Eric Weisstein's World of Mathematics, Carmichael Number
PROG
(PARI) is(n)=my(s); issquare(3*n+1, &s) && Mod(2, s-1)^(s-2)==1 && !isprime(s-1) && Mod(2, n)^n==2 && n>1 \\ Charles R Greathouse IV, Jul 07 2017
(PARI) forcomposite(p=1729, 1e6, n=p*(p+2)/3; if(Mod(2, p)^p==2 && Mod(2, n)^n==2, print1(n", "))) \\ Charles R Greathouse IV, Jul 07 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, Sep 03 2012
EXTENSIONS
a(3) and a(15) inserted by Charles R Greathouse IV, Jul 07 2017
STATUS
approved