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 2pi(x) > pi(2x) 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) (but see A212210-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

Crossrefs

Cf. A060207, A007097, A000720, A033844, A212210-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