User:IntegerSequences/Payam number

The Payam number $$E_\pm(e)$$ is the smallest positive odd integer $$k$$ such that for every positive integer $$n$$, the number  $$k\cdot2^n\pm1$$ is not divisible by any primes $$p$$ such that the multiplicative order of $$2\pmod{p}$$ is not larger than $$e$$.