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%I #26 Feb 24 2014 12:05:43
%S 0,1,2,4,2,2,4,1,2,3,3,6,2,9,4,6,7,4,4,6,3,3,3,5,2,6,4,4,4,8,3,4,6,3,
%T 3,8,8,6,6,7,7,10,7,6,14,5,5,8,5,4,6,3,3,13,2,14,12,12,12,18,18,18,11,
%U 11,11,17,10,11,11,16,16,9,9,16,15,16,8,14,14,14
%N Least number k among 1, ..., n such that pi(p+2*n) - pi(p+n) = pi(p+n) - pi(p) > 0 with p = prime(k), or 0 if such a number k does not exist, where pi(.) is given by A000720.
%C Conjecture: a(n) > 0 for all n > 1.
%H Zhi-Wei Sun, <a href="/A210210/b210210.txt">Table of n, a(n) for n = 1..5000</a>
%e a(4) = 4 since prime(4) = 7 with pi(7+2*4) - pi(7+4) = pi(7+4) - pi(7) = 1 > 0, but pi(2+2*4) - pi(2+4) = 1 < pi(2+4) - pi(2) = 2, pi(3+2*4) - pi(3+4) = 1 < pi(3+4) - pi(3) = 2, and pi(5+2*4) - pi(5+4) = 2 > pi(5+4) - pi(5) = 1.
%t p[k_,n_]:=Prime[k]+n>=Prime[k+1]&&k+PrimePi[Prime[k]+2n]==2*PrimePi[Prime[k]+n]
%t Do[Do[If[p[k,n],Print[n," ",k];Goto[aa]],{k,1,n}];
%t Print[n," ",0];Label[aa];Continue,{n,1,80}]
%Y Cf. A000040, A000720, A238281.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 24 2014
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