Given an integer e>=2, the Payam number E_±(e) is the smallest positive odd integer k such that for every positive integer n, the number k·2^n ± 1 is not divisible by any primes p such that the multiplicative order of 2 (mod p) is less than or equal to e. Payam numbers are good candidates for searching for Proth primes, i.e., primes of the form k·2^n + 1, as well as primes of the form k·2^n - 1. The first few values of E_+(n) for n = 2, 3, ... are 3, 9, 15, 105, 105, 105, 105, 105, 165, 165, 75075, ... (OEIS A083556), and the first few values of E_-(n) are 3, 3, 45, 45, 45, 45, 45, 45, 45, 2145, ... (OEIS A083391).
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