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A161865 Numerators of ratio of nonprimes in a square interval to that of nonprimes in that interval and its successor. 4
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 (list; graph; refs; listen; history; text; internal format)
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 A308180

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

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Last modified October 13 23:52 EDT 2019. Contains 327983 sequences. (Running on oeis4.)