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%I #8 Apr 17 2019 19:06:06
%S 1,1,9,3,3,25,7,21,435,241,3,153,151,493,537,2871,1713,4941,4963,307,
%T 28413,5035,1615,43525,9973
%N Smallest number k such that M(n)^2-k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).
%C All primes certified using openpfgw_v12 from primeform group
%e 3*3-1*3+1=7 prime 3=M(1)=2^2-1 so k(1)=1;
%e 7*7-1*7+1=43 prime 7=M(2)=2^3-1 so k(2)=1;
%e 31*31-9*31+1=683 prime 31=M(3)=2^5-1 so k(3)=9.
%t 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};
%t Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
%t While[! PrimeQ[m2 - k*m + 1], k++]; k, {n, 15}] (* _Robert Price_, Apr 17 2019 *)
%Y Cf. A000668, A139424, A139426, A139427, A139428, A139429, A139430, A139421.
%K hard,more,nonn
%O 1,3
%A _Pierre CAMI_, Apr 21 2008
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