A278792 a(n) is the positive integer x such that 3^((M-1)/(2*p)) == -2^x (mod M), where p > 2 is prime, M=2^p-1 is the n-th Mersenne prime and x < p.

2, 2, 1, 6, 16, 4, 5, 25, 18, 20, 45, 61, 91, 939, 817, 336, 862, 2533, 3404, 2822, 3136, 1554, 9371, 10712, 21311, 44296, 68185, 66909, 31147, 25648

Offset: 2

Author

Dmitry Ezhov, Nov 28 2016

Keywords

• nonn
• more

Comments

Let s=(M-1)/(2*p) and z is multiplicative order of 3 modulo M, then M|2^(x+p*i)+3^(s+z*j), where integer i,j>=0.

Crossrefs

Cf. A000043, A000668.

Programs

PARI

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];
for(n=2, #A000043, p= A000043[n]; M=2^p-1; s= (M-1)/2/p; x= valuation(lift(-Mod(3,M)^s), 2); print1(n,": ",x,", "));