%I #13 Feb 07 2014 04:10:47
%S 769,1729,2113,3025,2961,4481,6145,4321,6481,5625,7169,6841,8361,9289,
%T 12289,9729,11265,16129,16281,17065,24769,21761,21249,26641,34561,
%U 36289,34049,28081,32257,29745,32833,37889,43345,63361,38025,40609,72577,47433,71169
%N a(n) = 1+2*(d1 + 1)*(d2 + 1)* … *(dk + 1), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2 A001567(n).
%C It is notable how many primes, semiprimes, pseudoprimes, squares and multiples of 3 are in this sequence.
%C Primes obtained and the corresponding Fermat pseudoprime in the brackets: 769 (341), 2113 (645), 4481 (1729), 6481 (2465), 6841 (3277), 12289 (4371), 26641 (10585), 28081 (13747), 32257 (13981), 32833 (15709), 37889 (15841), 63361 (18705), 40609 (19951), 72577 (23001).
%C Semiprimes obtained and the corresponding Fermat pseudoprime in the brackets: 6145 (1905), 4321 (2047), 7169 (2821), 9289 (4369), 17065 (8321), 21761 (8911), 36289 (12801), 34049 (13741), 43345 (16705).
%C Pseudoprimes obtained and the corresponding Fermat pseudoprime in the brackets: 1729 (561).
%C Squares obtained and the corresponding Fermat pseudoprime in the brackets: 3025 = 5^2*11^2 (1105), 5625 = 3^2*5^4 (2701), 16129 = 127^2 (6601), 38025 = 3^2*5^2*13^2 (18721).
%C Multiples of 3 obtained and the corresponding Fermat pseudoprime in the brackets: 2961 = 3^2*329 (1387), 5625 = 3^2*625 (2701), 8361 = 3^2*929 (4033), 9729 = 3^2*1081 (4681), 3*3755 (5461), 16281 = 3^5*67 (7957), 21249 = 3^3*787 (10261), 29745 = 3^2*3305 (14491), 38025 = 3^2*4225 (18721), 47433 = 3*15811 (23377), 71169 = 3*23723 (25761).
%C The only numbers from the sequence above that are not into at least one of these categories (and the corresponding Fermat pseudoprime in the brackets) are 24769 = 17*31*47 (8481) and 34561 = 17*19*107 (11305).
%C An interesting correspondence with the function from the sequence A216404: with that one we obtain the pseudoprime 561 from the pseudoprime 1729 (2*a(n) + 1); with this one we obtain 1729 from 561 (a(n)). Another type of correspondence with that function: 2*a(n) + 1 = 769 for a(n) = 384 for that function (corresponding to pseudoprime 1905) while a(n) = 769 for this function (corresponding to pseudoprime 341).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%Y Cf. A001567, A216404.
%K nonn
%O 1,1
%A _Marius Coman_, Sep 12 2012