%I A120627
%S A120627 0,2,6,6,6,24,6,12,18,12,6,30,6,18,6,18,12,6,6,18,54,24,24,12,6,6,24,30,
%T A120627 42,18,12,18,30,12,24,6,36,18,6,54,84,30,36,18,30,12,30,54,6,42,18,12,
%U A120627 36,6,6,48,12,6,30,36,24,54,30,36,18,36,18,30,6,24,48,30,6,24,30,18,30
%N A120627 Least positive k such that both prime(n)+k and prime(n)+2k are prime, 
               or 0 if no such k exists.
%C A120627 Note that 6 divides a(n) for n>2. - T. D. Noe (noe(AT)sspectra.com), 
               Aug 29 2006
%C A120627 Van der Corput's theorem: There are infinitely many positive integers 
               n, k such that n, n+nk, n+2nk are all prime. - Jonathan Vos Post 
               (jvospost3(AT)gmail.com), Apr 17 2007
%D A120627 A. G. van der Corput, Uber Summen von Primzahlen und Primzahlquadraten,
               " Math. Ann., 116 (1939) 1-50.
%H A120627 T. D. Noe, <a href="b120627.txt">Table of n, a(n) for n=1..10000</a>
%H A120627 Terence Tao, <a href="http://terrytao.wordpress.com/2007/04/05/simons-lecture-i-structure-and-randomness-in-f\
               ourier-analysis-and-number-theory/">Simons Lecture I: Structure and 
               randomness in Fourier analysis and number theory</a>, April 2007.
%e A120627 a(3)=6 because prime(3)=5 and 5+6 and 5+12 are primes.
%t A120627 f[n_] := Block[{p = Prime[n], k = 1}, If[n == 1, 0, While[ ! PrimeQ[p 
               + 2k] || ! PrimeQ[p + 4k], k++ ]; 2k] ]; Table[f[n], {n, 80}] (*Chandler*)
%t A120627 Join[{0}, Table[p=Prime[n]; k=2; While[ !PrimeQ[p+k] || !PrimeQ[p+2k], 
               k=k+2]; k, {n,2,100}]] - T. D. Noe (noe(AT)sspectra.com), Aug 29 
               2006
%Y A120627 Cf. A000040.
%Y A120627 Sequence in context: A071888 A117217 A161331 this_sequence A089879 A087651 
               A078579
%Y A120627 Adjacent sequences: A120624 A120625 A120626 this_sequence A120628 A120629 
               A120630
%K A120627 easy,nonn
%O A120627 1,2
%A A120627 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Aug 25 2006
%E A120627 Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net) and 
               T. D. Noe (noe(AT)sspectra.com), Aug 28 2006