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%I #11 Mar 28 2015 17:09:14
%S 1105,1387,1729,2701,2821,4033,4681,5461,6601,8911,10261,10585,11305,
%T 13741,13981,14491,15841,16705,18721,29341,30121,30889,31609,31621,
%U 39865,41041,41665,46657,49141,52633,57421,63973,65281,68101,75361
%N Fermat pseudoprimes to base 2 of the form (6*k + 1)*(6*k*n + 1), where k, n are integers different from 0.
%C These are also called Poulet numbers. A few examples of how the formula looks like for k and n from 1 to 4:
%C For k = 1 the formula becomes 42*n + 7.
%C For k = 2 the formula becomes 156*n + 13.
%C For k = 3 the formula becomes 342*n + 19.
%C For k = 4 the formula becomes 600*n + 25.
%C For n = 1 the formula generates a perfect square.
%C For n = 2 the formula becomes (6*k + 1)*(12*k + 1) and were found the following Poulet numbers: 2701, 8911, 10585, 18721, 49141 etc.
%C For n = 3 the formula becomes (6*k + 1)*(18*k + 1) and were found the following Poulet numbers: 2821, 4033, 5461, 15841, 31621, 68101, etc.
%C For n = 4 the formula becomes (6*k + 1)*(24*k + 1). See the sequence A182123.
%C Note: the formula is equivalent to Poulet numbers of the form p*(n*p - n + 1), where p is of the form 6*k + 1. From the first 68 Poulet numbers just 7 of them (7957, 23377, 33153, 35333, 42799, 49981, 60787) can't be written as p*(n*p - n + 1), where p is of the form 6*k +- 1 and k, n are integers different from 0.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%t t = Select[Union[Flatten[Table[(6*k + 1)*(6*k*n + 1), {k, 100}, {n, 2000}]]], # < 76000 &]; Select[t, PowerMod[2, #, #] == 2 &] (* _T. D. Noe_, Jul 24 2012 *)
%Y Cf. A001567, A182123.
%K nonn
%O 1,1
%A _Marius Coman_, Jul 22 2012
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