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.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 14 2014

Keywords

• nonn
• look

Comments

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

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

Programs

Mathematica

tw[0]:=0
tw[n_]:=tw[n-1]+If[PrimeQ[Prime[n]+2],1,0]
SQ[n_]:=IntegerQ[Sqrt[tw[PrimePi[n]]]]
Do[Do[If[SQ[k*n-2],Print[n," ",k];Goto[aa]],{k,1,n}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]