For specific number theory question related to Mersenne numbers.

Mersenne numbers are numbers of the form of $M_n = 2^n -1$. Mersenne numbers are sometimes defined to have the additional requirement that n be prime, equivalently that they be pernicious Mersenne numbers, namely those numbers whose binary representation contains a prime number of ones and no zeros. The smallest composite pernicious Mersenne number is $2^{11} − 1 = 2047 = 23 \times 89$.

Mersenne prime is a mersenne numbers which is a prime number. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The first four Mersenne primes (sequence $A000668$ in the OEIS) are $3, 7, 31$, and $127$.

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