A216276 Fermat pseudoprimes to base 2 of the form (p^2 + 2*p)/3, where p is also a Fermat pseudoprime to base 2.

997633, 1398101, 2433601, 3581761, 26474581, 37354465, 63002501, 70006021, 82268033, 93030145, 561481921, 804978721, 1231726981, 2602378721, 2942952481, 12817618945, 15516020833, 16627811905, 22016333333, 25862624705, 53707855201, 67220090785, 95074073281, 144278347201

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

Cf. A001567, A216170.

Sequence in context: A055617 A055618 A341551 * A252903 A289140 A033426

Adjacent sequences: A216273 A216274 A216275 * A216277 A216278 A216279

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