A078435 -id:A078435 - cp's OEIS frontend

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

Daniel Tisdale, Jun 20 2009

			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.

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

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

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

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 *)

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.

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

A161182 Successive differences between positions of squares in list of nonprimes.

Original entry on oeis.org

1, 1, 3, 5, 6, 9, 9, 12, 13, 16, 16, 19, 20, 22, 25, 25, 26, 30, 31, 33, 34, 36, 38, 41, 40, 43, 46, 47, 48, 51, 53, 53, 56, 57, 60, 61, 64, 66, 65, 68, 69, 72, 76, 75, 78, 78, 83, 82, 82, 89, 90, 88, 89, 95, 96, 100, 101, 98, 104, 103, 105, 110, 108, 112, 115, 115, 118, 120

Offset: 1

Daniel Tisdale, Jun 05 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A078435, sequence of positions of squares in sequence of nonprimes.

A038107 := proc(n) numtheory[pi]( n^2) ; end: A078435 := proc(n) n^2-A038107(n) ; end: A161182 := proc(n) A078435(n)-A078435(n-1) ; end: seq(A161182(n),n=1..100) ; # R. J. Mathar, Jun 22 2009

With[{nn=5000},Differences[Flatten[Position[Complement[Range[0,nn], Prime[ Range[ PrimePi[nn]]]],?(IntegerQ[Sqrt[#]]&)]]]] (* _Harvey P. Dale, Mar 03 2014 *)

a(n) = A078435(n) - A078435(n-1). - R. J. Mathar, Jun 22 2009

Corrected and extended by R. J. Mathar, Jun 22 2009

A161183 Terms which occur at least twice in A161182.

Original entry on oeis.org

1, 9, 16, 25, 53, 78, 82, 89, 115, 120, 130, 152, 161, 178, 196, 224, 233, 235, 250, 256, 265, 286, 291, 300, 301, 314, 315, 325, 347, 357, 360, 368, 381, 391, 397, 419, 428, 430, 456, 468, 479, 483, 484, 494, 495, 512, 528, 570, 577, 589, 591, 608, 610, 620

Offset: 1

Daniel Tisdale, Jun 05 2009

A161182 is not monotonic, so some terms (like 89) listed here are not direct neighbors in A161182. - R. J. Mathar, Jun 22 2009

Cf. A078435, A161182.

A038107 := proc(n) numtheory[pi]( n^2) ; end: A078435 := proc(n) n^2-A038107(n) ; end: A161182 := proc(n) A078435(n)-A078435(n-1) ; end: L := [] ; for n from 1 to 1200 do L := [op(L), A161182(n)] ; od: read("transforms3") ; L := sort(L) ; L := LISTDUPL(L,0) ; # R. J. Mathar, Jun 22 2009

f[n_] := 2n - 1 - PrimePi[n^2] + PrimePi[(n-1)^2];
Select[Split[Array[f, 1000]//Sort], Length[#] >= 2&][[All, 1]]
(* Jean-François Alcover, Mar 07 2023 *)

Edited by N. J. A. Sloane, Jun 07 2009

Missing numbers added by R. J. Mathar, Jun 22 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A161182 Successive differences between positions of squares in list of nonprimes.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A161183 Terms which occur at least twice in A161182.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Extensions