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A237975
Least nonnegative integer m such that for some k = 1, ..., n there are exactly m^2 twin prime pairs not exceeding k*n.
2
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
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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 16 2014
STATUS
approved