A365160 Least k such that A000668(n) + k is prime, where A000668(n) is the n-th Mersenne prime.

2, 4, 6, 4, 18, 30, 22, 12, 16, 30, 40, 30, 888, 486, 2056, 696, 310, 718, 4692, 1600, 2788, 4290, 4326, 4150, 18088, 22096, 16342, 72816, 181720, 4200, 58416

Offset: 1

Author

Robert P. P. McKone, Aug 24 2023

Keywords

• nonn
• hard
• more

Comments

The distance between the n-th Mersenne prime and the next prime.

Examples

			A000668(6) = 131071, the next prime is 131101, so a(6) = 131101 - 131071 = 30.
		

Crossrefs

Cf. A000040, A000668 (Mersenne primes), A001223, A059305, A074626, A365161.

Programs

Mathematica

m[n_] := m[n] = (2^MersennePrimeExponent[n] - 1); a[k_, n_] := a[k, n] = m[n] + k; l[k_, n_] := l[k, n] = PrimeQ[a[k, n]]; Table[k = 1; Monitor[Parallelize[While[True, If[l[k, n], Break[]]; k++]; k], {n, k}], {n, 1, 20}]

Formula

a(n) = A001223(A059305(n)). - Michel Marcus, Aug 25 2023

a(n) = A074626(n) - A000668(n). - Amiram Eldar, Aug 10 2024

Extensions

a(28) from Michael S. Branicky, Aug 11 2024

a(29) from Minfeng Wang, Oct 29 2024

a(30)-a(31) from Minfeng Wang, Oct 29 2024