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

%I A216276 #22 Feb 16 2025 08:33:18
%S A216276 997633,1398101,2433601,3581761,26474581,37354465,63002501,70006021,
%T A216276 82268033,93030145,561481921,804978721,1231726981,2602378721,
%U A216276 2942952481,12817618945,15516020833,16627811905,22016333333,25862624705,53707855201,67220090785,95074073281,144278347201
%N A216276 Fermat pseudoprimes to base 2 of the form (p^2 + 2*p)/3, where p is also a Fermat pseudoprime to base 2.
%C A216276 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.
%C A216276 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.
%C A216276 Conjecture: For any Carmichael number c there exist infinitely many Carmichael numbers of the form (c^n + n*c)/(n + 1) with n > 1.
%H A216276 Charles R Greathouse IV, <a href="/A216276/b216276.txt">Table of n, a(n) for n = 1..10000</a>
%H A216276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%H A216276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%o A216276 (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
%o A216276 (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
%Y A216276 Cf. A001567, A216170.
%K A216276 nonn
%O A216276 1,1
%A A216276 _Marius Coman_, Sep 03 2012
%E A216276 a(3) and a(15) inserted by _Charles R Greathouse IV_, Jul 07 2017