A216646 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).

769, 1729, 2113, 3025, 2961, 4481, 6145, 4321, 6481, 5625, 7169, 6841, 8361, 9289, 12289, 9729, 11265, 16129, 16281, 17065, 24769, 21761, 21249, 26641, 34561, 36289, 34049, 28081, 32257, 29745, 32833, 37889, 43345, 63361, 38025, 40609, 72577, 47433, 71169

Offset: 1

Author

Marius Coman, Sep 12 2012

Keywords

• nonn

Comments

It is notable how many primes, semiprimes, pseudoprimes, squares and multiples of 3 are in this sequence.

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).

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).

Pseudoprimes obtained and the corresponding Fermat pseudoprime in the brackets: 1729 (561).

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).

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).

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).

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).

Links

• Eric Weisstein's World of Mathematics, Poulet Number

Crossrefs

Cf. A001567, A216404.