A319908 Lesser of twin primes pair p, such that the Mersenne numbers 2^p - 1 and 2^(p+2) - 1 have the same number of prime factors.

3, 5, 17, 71, 101, 137, 197, 269, 617, 857, 1019, 1049, 1061

Offset: 1

Author

Amiram Eldar, Oct 01 2018

Keywords

• nonn
• more

Comments

The corresponding number of prime factors is 1, 1, 1, 3, 2, 2, 2, 2, 4, 4, 5, 4, 2, ...

Assuming that Mersenne numbers (2^p-1 with p prime) are always squarefree, the distinction between number of prime factors with multiplicity (A001222) and number of different prime factors (A001221) is inessential.

Examples

			3 is in the sequence since 3 and 5 are twin primes pair, and 2^3-1=7 and 2^5-1=31 are both primes, thus having the same number of prime factors.
71 is in the sequence since 71 and 73 are twin primes pair and 2^71-1 and 2^73-1 both have 3 prime factors.
		

Crossrefs

Cf. A000225, A001359, A065341, A135975.

Programs

Mathematica

Do[If[PrimeQ[n]&&PrimeQ[n+2]&&PrimeOmega[2^n-1]==PrimeOmega[2^(n+2)-1],Print[n]],{n,1,200}]

PARI

isok(p) = isprime(p) && isprime(p+2) && (omega(2^p-1) == omega(2^(p+2)-1)); \\ Michel Marcus, Oct 02 2018