A214607 Fermat pseudoprimes to base 2 of the form (6*k + 1)*(6*k*n + 1), where k, n are integers different from 0.

1105, 1387, 1729, 2701, 2821, 4033, 4681, 5461, 6601, 8911, 10261, 10585, 11305, 13741, 13981, 14491, 15841, 16705, 18721, 29341, 30121, 30889, 31609, 31621, 39865, 41041, 41665, 46657, 49141, 52633, 57421, 63973, 65281, 68101, 75361

Offset

1,1

Comments

These are also called Poulet numbers. A few examples of how the formula looks like for k and n from 1 to 4:

For k = 1 the formula becomes 42*n + 7.

For k = 2 the formula becomes 156*n + 13.

For k = 3 the formula becomes 342*n + 19.

For k = 4 the formula becomes 600*n + 25.

For n = 1 the formula generates a perfect square.

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.

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.

For n = 4 the formula becomes (6*k + 1)*(24*k + 1). See the sequence A182123.

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.

Links

Table of n, a(n) for n=1..35.

Eric Weisstein's World of Mathematics, Poulet Number

Mathematica

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 *)

Crossrefs

Cf. A001567, A182123.

Sequence in context: A343084 A168629 A175521 * A321870 A257759 A025295

Adjacent sequences: A214604 A214605 A214606 * A214608 A214609 A214610

Keyword

nonn

Author

Marius Coman, Jul 22 2012

Status

approved