A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 12, 11, 12, 13, 13, 14, 14, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 15, 15, 15, 15, 15, 16, 15, 16

Offset: 1

Author

Cino Hilliard, Jul 22 2005

Keywords

• nonn
• easy

Comments

a(n) < log(4)*n/log(n) < 7*n/(5*log(n)) for n > 1. - Reinhard Zumkeller, Mar 04 2008

Bertrand's postulate is equivalent to the formula a(n) >= 1 for all positive integers n. - Jonathan Vos Post, Jul 30 2008

Number of distinct prime factors > n of binomial(2*n,n). - T. D. Noe, Aug 18 2011

f(2, 2n) - f(3, n) < a(n) < f(3, 2n) - f(2, n) for n > 5889 where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). The constant 3 can be improved. - Charles R Greathouse IV, May 02 2012

For n >= 2, a(n) is the number of primes appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

References

• F. Irschebeck, Einladung zur Zahlentheorie, BI Wissenschaftsverlag 1992, p. 40.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 181-182.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.

Crossrefs

Cf. A000720, A060715.

Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990, A074990.

Related sequences:

Primes (p) and composites (c): A000040, A002808, A000720, A065855.

Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.

Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

Programs

Maple

A108954 := proc(n)
    numtheory[pi](2*n)-numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 2017

Mathematica

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # > n &]], {n, 100}] (* T. D. Noe, Aug 18 2011 *)
f[n_] := Length@ Select[ Range[n + 1, 2n], PrimeQ]; Array[f, 100] (* Robert G. Wilson v, Mar 20 2012 *)
Table[PrimePi[2n]-PrimePi[n],{n,90}] (* Harvey P. Dale, Mar 11 2013 *)

PARI

g(n) = for(x=1,n,y=primepi(2*x)-primepi(x);print1(y","))

Python

from sympy import primepi
def A108954(n): return primepi(n<<1)-primepi(n) # Chai Wah Wu, Aug 19 2024

Formula

a(n) = A000720(2*n)-A000720(n).

For n > 1, a(n) = A060715(n). - David Wasserman, Nov 04 2005

Conjecture: G.f.: Sum_{i>0} Sum_{j>=i|i+j is prime} x^j. - Benedict W. J. Irwin, Mar 31 2017

From Wesley Ivan Hurt, Sep 20 2021: (Start)

a(n) = Sum_{k=1..n} A010051(2*n-k+1).

a(n) = Sum_{k=n*(n+1)/2+2..(n+1)*(n+2)/2} A010051(A128076(k)). (End)