Find factors with k = 2^n - 1 = Mersenne prime

[Dobri]

Examples:

n = 2, p = 5, k = 2ⁿ - 1 = M2, 2kp + 1 = 31 = M5;

n = 3, p = 929, k = 2ⁿ - 1 = M3, 2kp + 1 = 13007 is a factor of M929;

n = 5, p = 1973, k = 2ⁿ - 1 = M5, 2kp + 1 = 122327 is a factor of M1973;

n = 7, p = 37097, k = 2ⁿ - 1 = M7, 2kp + 1 = 9422639 is a factor of M37097;

n = 13, p = 1946669, k = 2ⁿ - 1 = M13, 2kp + 1 = 31890331559 is a factor of M1946669;

etc.

[Dobri]

Find also factors with k = 2^m(2ⁿ - 1), for example:

m = 7, n = 5, p = 270601, k = 2⁷(2ⁿ - 1) = 2⁷M5, 2kp + 1 = 2147489537 is a factor of M270601.

[Dobri]

Find also factors with k given as a product of Mersenne primes (not necessarily distinct ones) k = 2^m(2ⁿ¹ - 1)(2ⁿ² - 1)...

The challenge is to find a NEW factor.

[Dobri]

Let's set an upper limit for finding NEW factors, p < 10000001113, where
n = 2, p = 10000001113, k = 2ⁿ - 1 = M2, 2kp + 1 = 60000006679.

The leading entry should have the biggest n value. For starters, n = 2.

[Dobri]

Note that if there are several entries for a given n value, the corresponding leading entry for said n is the one with the smallest p value.