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.

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

A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().

Original entry on oeis.org

-1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 1, 1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3
Offset: 1

Views

Author

N. J. A. Sloane, May 04 2012

Keywords

Comments

It is conjectured that pi(x)+pi(y) >= pi(x+y) for 1 < y <= x.
A006093 gives row numbers of rows containing at least one negative term. [Reinhard Zumkeller, May 05 2012]

Examples

			Triangle begins:
  -1
  -1 0
   0 0 1
  -1 0 0 0
   0 0 1 1 2
  -1 0 1 1 1 1
   0 1 2 1 2 1 2
   0 1 1 1 1 1 2 2
   0 0 1 0 1 1 2 1 1
  -1 0 0 0 1 1 1 1 0 0
  ...

References

  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

Links

Crossrefs

Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.
Left diagonal is -A010051.

Programs

  • Haskell
    import Data.List (inits, tails)
a212210 n k = a212210_tabl !! (n-1) !! (k-1)
a212210_row n = a212210_tabl !! (n-1)
a212210_tabl = f $ tail $ zip (inits pis) (tails pis) where
   f ((xs,ys) : zss) = (zipWith (-) (map (+ last xs) (xs)) ys) : f zss
   pis = a000720_list
-- Reinhard Zumkeller, May 04 2012
  • Mathematica
    t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.