A035250 Number of primes between n and 2n (inclusive).

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

Offset

1,2

Comments

By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e., a(n) is positive for all n.

The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy, Jan 01 2007

The number of partitions of 2n into exactly two parts with first part prime, n > 1. - Wesley Ivan Hurt, Jun 15 2013

References

Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

International Mathematics Olympiad, Proof of Bertrand's Postulate [Via Wayback Machine]

Formula

a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - Reinhard Zumkeller, Jul 05 2010

a(A059316(n)) = n and a(m) <> n for m < A059316(n). - Reinhard Zumkeller, Jan 08 2012

a(n) = sum(A010051(k): k=n..2*n). [Reinhard Zumkeller, Jan 08 2012]

a(n) = pi(2n) - pi(n-1). [Wesley Ivan Hurt, Jun 15 2013]

Example

The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
a(5) = 2, since 2(5) = 10 has 5 partitions into exactly two parts: (9,1),(8,2),(7,3),(6,4),(5,5).  Two primes are among the first parts: 7 and 5.

Maple

with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

Mathematica

f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* Robert G. Wilson v, Dec 23 2012 *)

Prog

(Haskell)
a035250 n = sum $ map a010051 [n..2*n] -- Reinhard Zumkeller, Jan 08 2012
(Magma) [#PrimesInInterval(n, 2*n): n in [1..80]]; // Bruno Berselli, Sep 05 2012
(PARI) a(n)=primepi(2*n)-primepi(n-1) \\ Charles R Greathouse IV, Jul 01 2013

Crossrefs

Cf. A073837, A073838, A099802, A060715.

Sequence in context: A257212 A001031 A336543 * A165054 A067743 A029230

Adjacent sequences: A035247 A035248 A035249 * A035251 A035252 A035253

Keyword

nonn

Author

Erich Friedman

Status

approved