A382119 Numbers k = x*y such that (x*2^k - 1)*(y*2^k - 1) is semiprime.

2, 3, 4, 6, 16, 126

Offset: 1

Author

Juri-Stepan Gerasimov, Mar 25 2025

Keywords

• nonn
• more

Comments

The corresponding semiprimes are 21, 161, 961, 24257, 68718821377, 911862702743865038960621506943417274338971540779454191779448636691549662478337, ...

The pairs of primes corresponding to these semiprimes are [1st Mersenne prime, 2nd Mersenne prime], [2nd Mersenne prime, 23], [3rd Mersenne prime, 3rd Mersenne prime], [5th Mersenne prime, 191], [6th Mersenne prime, 7th Mersenne prime], [170141183460469231731687303715884105727, 5359447279004780799548150067050349330431], ...

a(7) > 100000. - Michael S. Branicky, Mar 31 2025

Examples

			2 = 1*2, 3 = 1*3, 4 = 2*2, 6 = 2*3, 16 = 2*8, 126 = 2*63.
		

Crossrefs

Cf. A000668, A001358, A161904.

Programs

Magma

[n: n in [1..1000] | not  #[d: d in Divisors(n) | IsPrime(d*2^n-1) and IsPrime((n div d)*2^n-1)] eq 0];