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

-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

1,5

Comments

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

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274. Shows a(n)>0 for n>10.

E. Labos, Illustration

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

Example

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

Mathematica

f[n_] := 2 PrimePi[n] - PrimePi[2 n]; Array[f, 122] (* Robert G. Wilson v, Aug 12 2011 *)

Prog

(PARI) a(n)=2*primepi(n)-primepi(2*n) \\ Charles R Greathouse IV, Jul 02 2013
(Magma) [2*#PrimesUpTo(n) -#PrimesUpTo(2*n): n in [1..200]]; // G. C. Greubel, Aug 01 2024
(SageMath) [2*prime_pi(n) -prime_pi(2*n) for n in range(1, 201)] # G. C. Greubel, Aug 01 2024

Crossrefs

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

Sequence in context: A127249 A127251 A063251 * A004570 A178064 A145363

Adjacent sequences: A060205 A060206 A060207 * A060209 A060210 A060211

Keyword

sign

Author

Labos Elemer, Mar 19 2001

Extensions

Edited by N. J. A. Sloane, Jul 03 2013

Status

approved