login
Least k such that k * M(n) * M(n+1) + 1 is prime where M(n) = A000668(n).
3

%I #22 Jan 03 2024 11:06:23

%S 2,6,4,16,36,6,42,24,172,18,52,54,6,130,2488,1344,12420,5596,364,178,

%T 3382,10516,44328,30342,25770,47146

%N Least k such that k * M(n) * M(n+1) + 1 is prime where M(n) = A000668(n).

%e 2*(2^2-1)*(2^3-1) + 1 = 43 is prime, so a(1) = 2.

%e 2*(2^3-1)*(2^5-1) + 1 = 435 is composite, 4*(2^3-1)*(2^5-1) + 1 = 867 is composite, 6*(2^3-1)*(2^5-1) + 1 = 1303 is prime, so a(2) = 6.

%t f[n_] := Module[{k = 1}, While[! PrimeQ[k*n + 1], k++]; k]; f /@ Times @@@ Partition[2^MersennePrimeExponent[Range[15]] - 1, 2, 1] (* _Amiram Eldar_, Jul 24 2021 *)

%t Table[k=1;x=(2^MersennePrimeExponent[n]-1)*(2^MersennePrimeExponent[n+1]-1);Monitor[Parallelize[While[True, If[PrimeQ[k*x+1],Break[]];k++];k],k],{n,1,24}] (* _J.W.L. (Jan) Eerland_, Jan 01 2024 *)

%Y Cf. A000668 (Mersenne primes).

%K nonn,more

%O 1,1

%A _Pierre CAMI_, Oct 17 2004

%E a(15)-a(23) from _Amiram Eldar_, Jul 24 2021

%E a(24)-a(25) from _J.W.L. (Jan) Eerland_, Jan 02 2024

%E a(26) from _Daniel Suteu_, Jan 03 2024