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).