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Least k such that A000668(n) - k is prime, where A000668(n) is the n-th Mersenne prime.
2

%I #42 Oct 30 2024 22:01:15

%S 1,2,2,14,12,8,18,18,30,20,170,24,114,56,156,2510,1824,12,3980,3630,

%T 16902,284,7712,20022,12930,9698,16232,1058,256016,23712,26298

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

%C The distance between the n-th Mersenne prime and the previous prime.

%F a(n) = A001223(A059305(n)-1). - _Michel Marcus_, Aug 25 2023

%F a(n) = A000668(n) - A073715(n). - _Amiram Eldar_, Aug 10 2024

%e A000668(6) = 131071, the previous prime is 131063, so a(6) = 131071 - 131063 = 8.

%t 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}]

%Y Cf. A000040, A000668 (Mersenne primes), A001223, A059305, A073715, A365160.

%K nonn,hard,more

%O 1,2

%A _Robert P. P. McKone_, Aug 24 2023

%E a(29)-a(31) from _Michael S. Branicky_, Sep 01 2024