A237879 - OEIS

%I #19 Apr 06 2014 22:23:45

%S 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 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 9,9,9,16,46,8,8,8,8,8,8,65,14,52,7,7,3,3,3,3

%N 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 According to the conjecture in A237840, a(n) should be always positive.

%H Zhi-Wei Sun, <a href="/A237879/b237879.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e 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 tw[0]:=0

%t tw[n_]:=tw[n-1]+If[PrimeQ[Prime[n]+2],1,0]

%t SQ[n_]:=IntegerQ[Sqrt[tw[PrimePi[n]]]]

%t Do[Do[If[SQ[k*n-2],Print[n," ",k];Goto[aa]],{k,1,n}];

%t Print[n," ",0];Label[aa];Continue,{n,1,100}]

%Y Cf. A000290, A001359, A006512, A237598, A237612, A237840.

%K nonn,look

%O 1,7

%A _Zhi-Wei Sun_, Feb 14 2014