A161865 Numerators of ratio of nonprimes in a square interval to that of nonprimes in that interval and its successor.

1, 3, 5, 2, 1, 3, 12, 13, 1, 16, 19, 10, 22, 1, 25, 13, 30, 31, 33, 17, 18, 38, 41, 40, 43, 46, 47, 16, 51, 1, 53, 56, 19, 60, 61, 32, 66, 65, 68, 23, 18, 76, 25, 1, 78, 83, 1, 82, 89, 45, 88, 89, 95, 24, 100, 101, 49, 104, 103, 21, 55, 27, 112, 1, 115, 59, 1, 20, 21, 15, 64, 1

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

The limit of this sequence is 1/2, as can be shown by setting an increasing lower bound on the ratio of composites in successive square intervals.

Example

First few terms are 1/4, 3/8, 5/11, 2/5, 1/2, 3/7, 12/25, 13/29.
For n=1: there is 1 nonprime <= 1, 2 nonprimes <= 4, and 5 nonprimes <= 9. The ratio is (2 - 1)/(5 - 1) = 1/4.

Maple

A062298 := proc(n) n-numtheory[pi](n) ; end: A078435 := proc(n) A062298(n^2) ; end: A161865 := proc(n) r := [ A078435(n), A078435(n+1), A078435(n+2)] ; (r[2]-r[1])/(r[3]-r[1]) ; numer(%) ; end: seq(A161865(n), n=1..120) ; # R. J. Mathar, Sep 27 2009

Mathematica

Numerator[Table[((2 n + 1) - (PrimePi[(n + 1)^2] - PrimePi[n^2]))/((4 n + 4) - (PrimePi[(n + 2)^2] - PrimePi[n^2])), {n, 1, 40}]] (* corrected by G. C. Greubel, Dec 20 2016 *)

Crossrefs

Cf. A161621, A161622, A161867 (denominators for this sequence).

Sequence in context: A140735 A183206 A197521 * A145325 A282194 A138508

Adjacent sequences: A161862 A161863 A161864 * A161866 A161867 A161868

Keyword

nonn,frac

Author

Daniel Tisdale, Jun 20 2009

Extensions

Extended beyond a(8) by R. J. Mathar, Sep 27 2009

Status

approved