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%I #7 Feb 18 2019 09:05:08
%S 3,4,5,6,7,8,10,13,14,18,21,22,26,27,28,32,33,35,51,54,58,67,76,89,
%T 100,140,170,189,269,307,365,408,470,475,546,558,604,751,771,857,892,
%U 896,992,1031,1080,1181,1228,1289,1342,1465,1483,1491,1520,1525,1571,1620
%N a(n) = numbers n such that A125683(n) is prime.
%C A125683(n) = Numerator[ Sum[ (-1)^(k+1) * 1/(k(k+1)), {k,1,n} ]. Corresponding primes are listed in A125685(n) = A125683[ a(n) ] = {5,11,2,79,331,479,5297,70061,69203,8960447,45083347,...}.
%e A125683(n) begins {1,1,5,11,2,79,331,479,493,5297,2701,69071,70061,...}.
%e Thus a(1) = 3 because A125683(3) = 5 is prime but A125683(k) is not prime for k<3.
%e a(2)-a(6) = {4,5,6,7,8} because A125683(k) is prime for 3<k<9.
%t g=0;Do[g=g+(-1)^(n+1)*1/(n(n+1));f=Numerator[g];If[PrimeQ[f],Print[{n,f}]],{n,1,1342}]
%Y Cf. A125683, A125685.
%K hard,nonn
%O 1,1
%A _Alexander Adamchuk_, Nov 30 2006
%E More terms from _Amiram Eldar_, Feb 18 2019
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