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

1,7

Comments

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

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A112106 A010608 A086139 * A074804 A242465 A075074

Adjacent sequences: A237876 A237877 A237878 * A237880 A237881 A237882

Keyword

nonn,look

Author

Zhi-Wei Sun, Feb 14 2014

Status

approved