cp's OEIS Frontend

A237879 Least positive integer k <= n such that the number of twin prime pairs not exceeding k*n is a square, or 0 if such a number k does not exist.

%I A237879 #19 Apr 06 2014 22:23:45
%S A237879 1,1,1,1,1,1,3,3,3,2,2,2,2,2,2,15,14,6,1,1,1,1,1,1,1,1,1,1,1,1,17,17,
%T A237879 7,7,3,3,15,14,6,6,13,13,13,12,12,5,5,5,11,11,11,2,2,2,10,10,10,4,4,4,
%U A237879 9,9,9,16,46,8,8,8,8,8,8,65,14,52,7,7,3,3,3,3
%N A237879 Least positive integer k <= n such that the number of twin prime pairs not exceeding k*n is a square, or 0 if such a number k does not exist.
%C A237879 According to the conjecture in A237840, a(n) should be always positive.
%H A237879 Zhi-Wei Sun, <a href="/A237879/b237879.txt">Table of n, a(n) for n = 1..10000</a>
%H A237879 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A237879 a(7) = 3 since there are exactly 2^2 = 4 twin prime pairs not exceeding 3*7 = 21 (namely, {3, 5}, {5, 7}, {11, 13} and {17, 19}), but the number of twin prime pairs not exceeding 1*7 and the number of twin prime pairs not exceeding 2*7 are 2 and 3 respectively, none of which is a square.
%t A237879 tw[0]:=0
%t A237879 tw[n_]:=tw[n-1]+If[PrimeQ[Prime[n]+2],1,0]
%t A237879 SQ[n_]:=IntegerQ[Sqrt[tw[PrimePi[n]]]]
%t A237879 Do[Do[If[SQ[k*n-2],Print[n," ",k];Goto[aa]],{k,1,n}];
%t A237879 Print[n," ",0];Label[aa];Continue,{n,1,100}]
%Y A237879 Cf. A000290, A001359, A006512, A237598, A237612, A237840.
%K A237879 nonn,look
%O A237879 1,7
%A A237879 _Zhi-Wei Sun_, Feb 14 2014