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If M(n) is the n-th Mersenne prime, then a(n) is the smallest positive integer such that 2*a(n)*M(n)*M(n+1)*M(n+2)-1 is prime.
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%I #6 Dec 09 2018 17:22:21

%S 1,1,21,1,12,16,6,112,76,195,61,21,511,909,1689,517,640,487,13615,

%T 12547,382,60456

%N If M(n) is the n-th Mersenne prime, then a(n) is the smallest positive integer such that 2*a(n)*M(n)*M(n+1)*M(n+2)-1 is prime.

%e a(3) = 21: M(3) = 2^5-1 = 31; M(4) = 2^7-1 = 127; M(5) = 2^13-1 = 8191; 2*21*31*127*8191-1 = 1354414613, which is prime.

%t spi[n_]:=Module[{k=2},While[!PrimeQ[k*n-1],k+=2];k/2]; spi/@Times@@@ Partition[ Select[2^Range[5000]-1,PrimeQ],3,1] (* The program generates the first 18 terms of the sequence. To generate more terms, increase the Range specification constant, but the program may take a long time to run. *) (* _Harvey P. Dale_, Dec 09 2018 *)

%K nonn

%O 1,3

%A _Ray G. Opao_, Apr 21 2004