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%I #14 Jul 07 2017 11:05:10
%S 831405,5977153,15913261,21474181,38171953,126619741,210565981,
%T 224073865,327718401,377616421,390922741,558097345,699735345,
%U 700932961,1327232481,1999743661,4996150741,8523152641,11358485281,13999580785,15613830541,17657245081,20442723301
%N Fermat pseudoprimes to base 2 that can be written as 2*p^2 - p, where p is also a Fermat pseudoprime to base 2.
%C Fermat pseudoprimes are listed in A001567.
%C The correspondent p for the numbers from the sequence above: 645, 1729, 2821, 3277, 4369, 7957, 10261, 10585, 12801, 13741, 13981, 16705, 18705, 25761, 31621, 49981, 65281, 75361, 83665, 88357, 93961, 101101.
%C Note that for 22 of the first 80 Poulet numbers, we obtained through this formula another Poulet number!
%C The formula could be generalized this way: Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number.
%C For n = 1, that formula becomes the formula set out for the sequence above.
%C For n = 2, that formula becomes 3*p^2 - 2*p, from which the Poulet numbers 348161 (for p = 341) and 1246785 (for p = 645) were obtained.
%C For n = 3, that formula becomes 4*p^2 - 3*p, from which the Poulet number 119273701 (for p = 5461) was obtained.
%C For n = 4, that formula becomes 5*p^2 - 4*p, from which the Poulet numbers 2077545 (for p = 645) and 9613297 (for p = 1387) were obtained.
%C Conjecture: there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number.
%C Finally, considering, e.g., that for the Poulet number 645, Poulet numbers were obtained for n = 1, 2, 4 (i.e., 831405, 1246785, 2077545), yet another conjecture: For any Poulet number p, there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0.
%H Charles R Greathouse IV, <a href="/A215343/b215343.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%Y Cf. A001567, A213812.
%K nonn
%O 1,1
%A _Marius Coman_, Aug 08 2012
%E Edited by _Jon E. Schoenfield_, Dec 12 2013
%E a(14) inserted by _Charles R Greathouse IV_, Jul 07 2017
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