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 1-2 of 2 results.

A071682 Least m such that A060207(m) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 13, 16, 20, 24, 28, 32, 37, 42, 47, 52, 57, 62, 68, 73, 79, 85, 91, 97, 103, 109, 116, 122, 129, 135, 142, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 218
Offset: 2

Views

Author

David Wasserman, May 31 2002

Keywords

Links

Crossrefs

Cf. A060207, A007097.

Formula

a(n) = ceiling(log_2(A007097(n-2))).

A060208 a(n) = 2*pi(n) - pi(2*n), where pi(i) = A000720(i).

Original entry on oeis.org

-1, 0, 1, 0, 2, 1, 2, 2, 1, 0, 2, 1, 3, 3, 2, 1, 3, 3, 4, 4, 3, 2, 4, 3, 3, 3, 2, 2, 4, 3, 4, 4, 4, 3, 3, 2, 3, 3, 3, 2, 4, 3, 5, 5, 4, 4, 6, 6, 5, 5, 4, 3, 5, 4, 3, 3, 2, 2, 4, 4, 6, 6, 6, 5, 5, 4, 6, 6, 5, 4, 6, 6, 8, 8, 7, 6, 6, 6, 7, 7, 7, 6, 8, 7, 7, 7, 6, 6, 8, 7, 6, 6, 6, 6, 6, 5, 6, 6, 5, 4, 6, 6, 8, 8, 8, 7, 9, 9, 11, 11, 11, 10, 12, 11, 10, 10, 9, 9, 9, 8, 7, 7
Offset: 1

Views

Author

Labos Elemer, Mar 19 2001

Keywords

Comments

Rosser & Schoenfeld show 2*pi(x) > pi(2*x) for x > 10. - N. J. A. Sloane, Jul 03 2013, corrected Jul 09 2015

Examples

			n=100, pi(100)=25, pi(200)=46, 2pi(100)-pi(2*100) =4=a(100)

References

  • J. Barkley Rosser and Lowell Schoenfeld, Abstracts of Scientific Communications, Internat. Congress Math., Moscow, 1966, Section 3, Theory of Numbers.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.
  • Sanford Segal, On Pi(x+y)<=Pi(x)+Pi(y). Transactions American Mathematical Society, 104 (1962), 523-527.

Links

Crossrefs

Cf. A000720, A007097, A033844, A060207, A212210, A212211, A212212, A212213.

Programs

  • Magma
    [2*#PrimesUpTo(n) -#PrimesUpTo(2*n): n in [1..200]]; // G. C. Greubel, Aug 01 2024
  • Mathematica
    f[n_] := 2 PrimePi[n] - PrimePi[2 n]; Array[f, 122] (* Robert G. Wilson v, Aug 12 2011 *)
  • PARI
    a(n)=2*primepi(n)-primepi(2*n) \\ Charles R Greathouse IV, Jul 02 2013
  • SageMath
    [2*prime_pi(n) -prime_pi(2*n) for n in range(1,201)] # G. C. Greubel, Aug 01 2024

Formula

a(n) = Mod[2*PrimePi[n], PrimePi[2n]] = 2*A000720(n) - A000720(2n) for n>1.
a(n) ~ 2n log 2 / (log n)^2, by the prime number theorem. - N. J. A. Sloane, Mar 12 2007
a(n) = -A047886(n,n) (see A212210 to A212213). - Reinhard Zumkeller, Apr 15 2008

Extensions

Edited by N. J. A. Sloane, Jul 03 2013
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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