A237975 Least nonnegative integer m such that for some k = 1, ..., n there are exactly m^2 twin prime pairs not exceeding k*n.

0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 5, 4, 4, 3, 3, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 3, 3, 3, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6, 9, 5, 5, 5, 5, 5, 5, 11, 6, 10, 5, 5, 4, 4, 4, 4, 5, 11, 9, 8, 9, 6, 10, 5, 5, 5, 5, 5, 5, 5, 5, 8, 11, 11, 7, 8

Offset

1,7

Comments

The conjecture in A237840 implies that a(n) exists for any n > 0.

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) = 2 since there are exactly 2^2 twin prime pairs not exceeding 3*7  = 21 (namely, {3, 5}, {5, 7}, {11, 13} and{17,19}), and the number of twin prime pairs not exceeding 1*7 or 2*7 is not a square.
a(18055) = 675 since there are exactly 675^2 = 455625 twin prime pairs not exceeding 5758*18055.

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, " ", Sqrt[tw[PrimePi[k*n-2]]]]; Goto[aa]], {k, 1, n}]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000290, A001359, A006512, A237840, A237879.

Sequence in context: A348001 A361923 A032572 * A327343 A032574 A058515

Adjacent sequences: A237972 A237973 A237974 * A237976 A237977 A237978

Keyword

nonn

Author

Zhi-Wei Sun, Feb 16 2014

Status

approved