%I #13 Oct 25 2014 23:20:19
%S 0,10,26,30,36,136,156,433
%N Numbers n such that there is no prime of a prime twin pair between n^2 + n and n^2 + 3*n + 2.
%C No more values for n = 434 to 45140.
%C Conjecture: the sequence is finite and given in full.
%C a(9), if it exists, is greater than 10^5. - _Derek Orr_, Sep 19 2014
%e n = 0 only 1 between 0 and 2 so a(1) = 0.
%e n = 1 between 2 and 6, 3 is the first of twin pair 3, 5.
%e For n = 2 to 9 always at least one prime of a twin pair between n^2 + n and n^2 + 3*n + 2.
%e n = 10 no prime of a twin pair between 110 and 132 so a(2) = 10.
%o (PARI)
%o a(n)=forprime(p=n^2+n,n^2+3*n+2,if(precprime(p-1)==p-2||nextprime(p+1)==p+2,return(0)));return(1)
%o n=0;while(n<10^5,if(a(n),print1(n,", "));n++) \\ _Derek Orr_, Sep 19 2014
%Y Cf. A091592, A108309.
%K nonn
%O 1,2
%A _Pierre CAMI_, Sep 04 2014