Poulet Number

A Poulet number is a Fermat pseudoprime to base 2, denoted psp(2), i.e., a composite number  such that

The first few Poulet numbers are 341, 561, 645, 1105, 1387, ... (OEIS A001567).

Pomerance et al. (1980) computed all  Poulet numbers less than . The numbers less than , , ..., are 0, 3, 22, 78, 245, ... (OEIS A055550).

Pomerance has shown that the number of Poulet numbers less than  for sufficiently large  satisfy

(Guy 1994).

A Poulet number all of whose divisors  satisfy  is called a super-Poulet number. There are an infinite number of Poulet numbers which are not super-Poulet numbers. Shanks (1993) calls any integer satisfying  (i.e., not limited to odd composite numbers) a Fermatian.

REFERENCES:

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 28-29, 1994.

Pinch, R. G. E. "The Pseudoprimes Up to ." ftp://ftp.dpmms.cam.ac.uk/pub/PSP/.

Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The Pseudoprimes to ." Math. Comput. 35, 1003-1026, 1980. http://mpqs.free.fr/ThePseudoprimesTo25e9.pdf.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 115-117, 1993.

Sloane, N. J. A. Sequences A001567/M5441 and A055550 in "The On-Line Encyclopedia of Integer Sequences."