A139425 Smallest number k such that M(n)^2-k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

1, 1, 9, 3, 3, 25, 7, 21, 435, 241, 3, 153, 151, 493, 537, 2871, 1713, 4941, 4963, 307, 28413, 5035, 1615, 43525, 9973

Offset: 1

Author

Pierre CAMI, Apr 21 2008

Keywords

• hard
• more
• nonn

Comments

All primes certified using openpfgw_v12 from primeform group

Examples

			3*3-1*3+1=7 prime 3=M(1)=2^2-1 so k(1)=1;
7*7-1*7+1=43 prime 7=M(2)=2^3-1 so k(2)=1;
31*31-9*31+1=683 prime 31=M(3)=2^5-1 so k(3)=9.
		

Crossrefs

Cf. A000668, A139424, A139426, A139427, A139428, A139429, A139430, A139421.

Programs

Mathematica

A000043 = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 - k*m + 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)