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%I #9 Jul 23 2021 05:30:45
%S 1,3,5,8,99,275,278,404,96,1538,1253,15858,189168,119552,221444,
%T 1047122,3571449,5424924,1575995
%N Least number k such that (k*M(n))^2 + k*M(n) - 1 and (k*M(n))^2 + k*M(n) + 1 are twin primes where M(n) is the n-th Mersenne prime.
%e M(2) = 2^3-1 = 7, 7^2+7-1 = 55 is composite, (2*7)^2+2*7-1 = 209 is composite,
%e (3*7)^2+3*7-1 = 461 is prime, 461 and 463 are twin primes, so a(2) = 3.
%t f[p_] := Module[{k = 1}, While[!PrimeQ[(k*p)^2 + k*p - 1] || !PrimeQ[(k*p)^2 + k*p + 1], k++]; k]; f /@ (2^MersennePrimeExponent[Range[10]] - 1) (* _Amiram Eldar_, Jul 23 2021 *)
%Y Cf. A000043, A000668, A001097, A121370.
%K nonn,hard,more
%O 1,2
%A _Pierre CAMI_, Jul 24 2006
%E a(8) corrected and a(18)-a(19) added by _Amiram Eldar_, Jul 23 2021
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