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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. 0

%I

%S 2,2,1,6,16,4,5,25,18,20,45,61,91,939,817,336,862,2533,3404,2822,3136,

%T 1554,9371,10712,21311,44296,68185,66909,31147,25648

%N 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.

%C 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.

%o (PARI)

%o 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];

%o 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,", "));

%Y Cf. A000043, A000668.

%K nonn,more

%O 2,1

%A _Dmitry Ezhov_, Nov 28 2016

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Last modified August 20 06:04 EDT 2019. Contains 326139 sequences. (Running on oeis4.)