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%I #16 Sep 05 2013 16:28:57
%S 2,3,7,5,17,14,16,24,12,19,28,43,86,80,34,82,78,73,69,66,117,329,57,
%T 222,171,228,178,470,291,359,505,366,585,576,644,544,423,742,502,636,
%U 765,466,936,578,697,682,541,1442,640,627,615,603,2025,1660,570,1833
%N a(n) is the maximal k>=1 such that nextprime(j*n)<=(j+1)*n, j=1,...,k.
%F Conjectural inequality: for n>=2, a(n) <= log^2(n*a(n)). This essentially corresponds to Cramer's conjecture for prime gaps.
%e If n=3, then, for j=1, nextprime(3)<=6; for j=2, nextprime(6)<=9; for j=3,nextprime(9)<=12; for j=4, nextprime(12)<=15; for j=5, nextprime(15)<=18; for j=6,nextprime(18)<=21; for j=7, nextprime(21)<=24, BUT for j=8, nextprime(24)>27. Thus a(3)=7.
%t a[n_] := For[k = 1, True, k++, If[NextPrime[k*n] <= (k+1)*n && NextPrime[(k+1)*n] > (k+2)*n, Return[k]]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Sep 05 2013 *)
%Y Cf. A002386, A005250, A111870.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Sep 04 2013
%E More terms from _Peter J. C. Moses_
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