A121370 Least number k such that (k*M(n))^2 + k*M(n) - 1 is prime with M(i)=i-th Mersenne prime.

1, 3, 1, 7, 8, 19, 13, 4, 16, 3, 42, 24, 434, 84, 160, 579, 475, 529, 2450, 2644, 3928, 558, 13680, 7146, 1408, 3003, 2369, 55000, 83873

Offset: 1

Author

Pierre CAMI, Jul 24 2006

Keywords

• hard
• more
• nonn

Examples

			M(4)=2^7-1=127
127^2+127-1=16255 composite
(2*127)^2+2*127-1=64769 composite
(3*127)^2+3*127-1=145541 composite
(4*127)^2+4*127-1=258571 composite
(5*127)^2+5*127-1=403859 composite
(6*127)^2+6*127-1=581405 composite
(7*127)^2+7*127-1=791209 prime so k(4)=7
1*(2^2-1)*(1*(2^2-1)+1)-1=11 prime, 2^2-1 first Mersenne prime, a(1)=1.
3*(2^3-1)*(3*(2^3-1)+1)-1=461 prime, 2^3-1 second Mersenne prime, a(2)=3.
n=6: Mp(6) = 131071 and 19*131071*(19*131071 + 1) - 1 = 6201840632149 which is prime, and for k=1..18 no prime appears. - _Wolfdieter Lang_, Oct 26 2014
		

Crossrefs

Cf. A121371.

Cf. A000043 (Mersenne exponents), A000668 (Mersenne primes).

Cf. A137906, A137907, A137909.

Programs

PARI

lista() = {v = readvec("b000043.txt"); for (i=1, #v, mp = 2^v[i] - 1; k=1; while (!isprime(k*mp*(k*mp + 1) - 1), k++); print1(k, ", "););} \\ Michel Marcus, Oct 27 2014

Formula

a(n) is the least k >= 1 for which k*Mp(n)*(k*Mp(n) + 1) - 1 is prime, where Mp(n) = A000668(n) (see Name). - Wolfdieter Lang, Oct 26 2014

Extensions

a(21) corrected by Pierre CAMI, Mar 04 2014

a(27)-a(29) by Pierre CAMI, Oct 11 2014

Checked for n = 1..15 by Wolfdieter Lang, Oct 26 2014

Merged with A137908 by Vaclav Kotesovec, Oct 30 2014