A319908 - OEIS

%I #14 Nov 20 2018 05:19:51

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

%N 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.

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

%C 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.

%e 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.

%e 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.

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

%o (PARI) isok(p) = isprime(p) && isprime(p+2) && (omega(2^p-1) == omega(2^(p+2)-1)); \\ _Michel Marcus_, Oct 02 2018

%Y Cf. A000225, A001359, A065341, A135975.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Oct 01 2018