cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A161621 Numerator of (b(n+1) - b(n))/(b(n+2) - b(n)), where b(n) = A038107(n) is the number of primes up to n^2.

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 3, 4, 3, 5, 4, 1, 5, 2, 6, 7, 5, 1, 6, 1, 1, 7, 2, 9, 8, 7, 8, 9, 1, 4, 10, 9, 10, 9, 10, 1, 3, 12, 11, 12, 11, 3, 12, 11, 13, 10, 13, 3, 10, 11, 15, 4, 12, 13, 11, 12, 17, 13, 1, 16, 13, 17, 15, 7, 16, 1, 15, 17, 13, 7, 1, 15, 1, 17, 9, 11, 7, 18, 23, 13, 20, 19, 20, 17, 16
Offset: 1

Views

Author

Daniel Tisdale, Jun 14 2009

Keywords

Comments

If the limit of R(n) exists as n->oo it is 1/2, but existence of the limit is conjectural. R(n) generalizes to R_k(n) by substituting PrimePi_k for PrimePi(n), where PrimePi_k(n) is the number of numbers with k prime factors (including repetitions) <= n. Convergence of {R(n)} to 1/2 implies Legendre's conjecture. For discussion of the order of the number of prime factors of a number n see reference [1], below. The PNT and reference [1] suggest but offer no proof that R_k(n)-> 1/2 as n -> oo. The corresponding sequence for near-primes would be {R_2(n)} = {1/3, 2/3, 1/2, ...}.

Examples

			R(3) = (PrimePi(4^2)-PrimePi(3^2)) / (PrimePi(5^2)-PrimePi(3^2)) = (PrimePi(16)-PrimePi(9)) / (PrimePi(25)-PrimePi(9)) = (6-4)/(9-4) = 2/5. Hence a(3) = 2. - _Klaus Brockhaus_, Jun 15 2009

References

  • S. Ramanujan, The Normal Number of Prime Factors of a Number n, reprinted at Chapter 35, Collected Papers (Hardy et al., ed), AMS Chelsea Publishing, 2000.

Links

Crossrefs

Cf. A014085
Cf. A161622 (denominators). - Klaus Brockhaus, Jun 15 2009

Programs

  • Magma
    [ Numerator((#PrimesUpTo((n+1)^2)-a) / (#PrimesUpTo((n+2)^2)-a)) where a is #PrimesUpTo(n^2): n in [1..85] ]; // Klaus Brockhaus, Jun 15 2009
  • Mathematica
    Numerator[(#[[2]]-#[[1]])/(#[[3]]-#[[1]])&/@Partition[PrimePi[ Range[ 90]^2],3,1]] (* Harvey P. Dale, Jan 06 2017 *)

Extensions

a(1) inserted and extended beyond a(13) by Klaus Brockhaus, Jun 15 2009
Simplified title by John W. Nicholson, Dec 13 2013

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

Original entry on oeis.org

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

Views

Author

Daniel Tisdale, Jun 20 2009

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • 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 *)

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.

Extensions

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

A163384 Values of n for which the ratio A161865/A161867 is greater than 1/2.

Original entry on oeis.org

23, 37, 42, 46, 50, 56, 58, 61, 69, 75, 77, 78, 80, 85, 92, 105, 107, 110, 116, 119, 122, 129, 133, 137, 138, 140, 143, 145, 147, 149, 152, 153, 157, 159, 162, 165, 168, 172, 174, 175, 178, 180, 182, 184, 188, 190, 192, 195, 201, 203, 205, 210, 216, 219, 222, 225, 228, 229, 232
Offset: 1

Views

Author

Daniel Tisdale, Jul 25 2009

Keywords

Comments

The corresponding values of A161621/A161622 are less than 1/2. For given n, both sequences would not be simultaneously greater than 1/2, although both could be less than 1/2.

Links

Programs

  • Mathematica
    Select[Range[100], ((2 # + 1) - (PrimePi[(# + 1)^2] - PrimePi[#^2]))/((4 # + 4) - (PrimePi[(# + 2)^2] - PrimePi[#^2])) > 1/2 &] (* G. C. Greubel, Dec 20 2016 *)

Extensions

Added missing term 58 and terms a(14) onward by G. C. Greubel, Dec 20 2016

A163161 Values of n at which the sequence A161865/A161867 equals 1/2.

Original entry on oeis.org

5, 9, 14, 30, 44, 47, 64, 67, 72, 89, 99, 124, 125, 194, 198, 200, 208, 213, 214, 249, 267, 270, 320, 333, 372, 373, 374, 393, 455, 460, 475, 519, 586, 603, 643, 701, 704, 712, 770, 774, 789, 814, 825, 828, 836, 891, 906, 930, 941, 970, 974, 982, 983, 988, 1022
Offset: 1

Views

Author

Daniel Tisdale, Jul 21 2009, Jul 24 2009

Keywords

Comments

The corresponding sequence for primes is A144264, and the ratio is given by sequences A161621/A161622.

Programs

  • Mathematica
    t1 = Table[ ((2n+1) - (PrimePi[(n+1)^2] - PrimePi[n^2]))/((4n+4)- (PrimePi[(n+2)^2] - PrimePi[n^2])), {n,1,300}]; t2= Position[t1, 1/2] (* Daniel Tisdale, Jul 24 2009 *)

Extensions

Corrected by the author, Jul 25 2009
Extended beyond 194 by R. J. Mathar, Sep 27 2009
Showing 1-4 of 4 results.

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