%I #8 Nov 28 2014 21:55:08
%S 1,1,1,1,1,2,2,2,3,2,4,3,4,4,3,4,3,3,2,1,2,3,2,2,5,4,3,2,1,1,4,3,2,3,
%T 4,6,5,4,3,4,5,6,7,6,5,4,5,5,5,4,5,4,5,5,5,6,9,8,8,7,7,7,8,7,6,6,5,4,
%U 3,3,3,2,7,6,5,5,5,4,3,2,5,5,8,9,11,10,10,9,9,8,7,7,6,6,6,5,4,5,8,8
%N Number of primes p such that p+2 is prime and A001359(n) < p < A001359(n)^(1+1/n).
%C Conjecture: For every positive integer n, A001359(n+1)^(1/(n+1)) < A001359(n)^(1/n). Note that this conjecture is equivalent to " A001359 is infinite and for every n, A001359(n+1) < A001359(n)^(1+1/n). This implies for every n, a(n) is positive. See comment lines of the sequence A001359.
%e Take n=1, A001359(1)=3, 3 < 5 < 3^(1+1/1)=9 hence a(1)=1.
%e Take n=6, A001359(6)=41, 41 < 59 < 71 < 41^(1+1/6)~76.13 hence a(6)=2.
%Y Cf. A001359, A248901, A248902, A248903.
%K nonn
%O 1,6
%A _Jahangeer Kholdi_ and _Farideh Firoozbakht_, Nov 22 2014