Given an integer , the Payam number  is the smallest positive odd integer  such that for every positive integer , the number  is not divisible by any primes  such that the multiplicative order of 2  is less than or equal to . Payam numbers are good candidates for searching for Proth primes, i.e., primes of the form , as well as primes of the form .

The first few values of  for , 3, ... are 3, 9, 15, 105, 105, 105, 105, 105, 165, 165, 75075, ... (OEIS A083556), and the first few values of  are 3, 3, 45, 45, 45, 45, 45, 45, 45, 2145, ... (OEIS A083391).

REFERENCES:

Sloane, N. J. A. Sequences A083391 and A083556 in "The On-Line Encyclopedia of Integer Sequences."

Smith, R. "A Co-Ordinated Search for Primes in the Payam Number Series." http://home.btclick.com/rwsmith/pp/payam3.htm.