A255909 - OEIS

%I #20 Apr 15 2015 16:51:50

%S 1,4,2,2,2,6,2,2,4,4,4,2,6,2,2,4,2,8,6,6,2,2,2,2,2,20,6,6,2,2,8,2,4,4,

%T 2,6,2,4,4,4,4,4,4,4,12,10,2,12,12,6,8,4,4,2,8,16,10,2,18,6,6,6,2,2,4,

%U 8,2,18,2,14,4,4,4,4,10,10,4,2,10,4,4,2

%N Second difference sequence of A070865.

%C Does 2 occur infinitely many times?  If so, does every even positive integer occur infinitely many times?  More generally, suppose that p < q are primes, and let p(1) = p, p(2) = q, and, for n > 2, let p(n) = least prime h such that h - p(n-1) > p(n-1) - p(n-2).  Does every even positive integer occur infinitely many times in the second difference sequence of (p(n))?

%H Clark Kimberling, <a href="/A255909/b255909.txt">Table of n, a(n) for n = 1..10000</a>

%e A070865 = (2,3,5,11,19,29,41,59,79,...)

%e 1st differences:  1,2,6,8,10,12,18,20,...

%e 2nd differences:  1,4,2,2,2,6,2,...

%t d = 0; p = 2; t = {p}; Do[d = NextPrime[p + d] - p; AppendTo[t, p += d], {200}]; t

%t Differences[t,2]  (* A255909 *)

%t (* uses Vladimir Joseph Stephan Orlovsky's program at A070865 *)

%Y Cf. A070865, A000040.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 14 2015