A060715 Number of primes between n and 2n exclusive.

0, 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

Offset: 1

Author

Lekraj Beedassy, Apr 25 2001

Keywords

• nonn
• easy

Comments

See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008

a(A060756(n)) = n and a(m) <> n for m < A060756(n). - Reinhard Zumkeller, Jan 08 2012

For prime n conjecturally a(n) = A226859(n). - Vladimir Shevelev, Jun 27 2013

The number of partitions of 2n+2 into exactly two parts where the first part is a prime strictly less than 2n+1. - Wesley Ivan Hurt, Aug 21 2013

Examples

			a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.
		

References

• M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from T. D. Noe]
• C. K. Caldwell, The Prime Glossary, Bertrand's postulate
• R. Chapman, Bertrand postulate [Broken link]
• Math Olympiads, Bertrand's Postulate [Broken link]
• S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
• Vladimir Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, arXiv:0909.0715v13 [math.NT]
• Vladimir Shevelev, Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
• M. Slone, PlanetMath.org, Proof of Bertrand's conjecture
• Jonathan Sondow and Eric Weisstein, Bertrand's Postulate, World of Mathematics
• Wikipedia, Proof of Bertrand's postulate
• Dr. Wilkinson, The Math Forum, Erdos' Proof
• Wolfram Research, Bertrand hypothesis

Crossrefs

Cf. A060756, A070046, A006992, A051501, A035250, A101909.

Cf. A000720, A014085, A104272, A143227, A128076.

Cf. A143223, A143224, A143225, A143226.

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

Haskell

a060715 n = sum $ map a010051 [n+1..2*n-1]  -- Reinhard Zumkeller, Jan 08 2012

Magma

[0] cat [#PrimesInInterval(n+1, 2*n-1): n in [2..80]]; // Bruno Berselli, Sep 05 2012

Maple

a := proc(n) local counter, i; counter := 0; from i from n+1 to 2*n-1 do if isprime(i) then counter := counter +1; fi; od; return counter; end:
with(numtheory); seq(pi(2*k-1)-pi(k),k=1..100); # Wesley Ivan Hurt, Aug 21 2013

Mathematica

a[n_]:=PrimePi[2n-1]-PrimePi[n]; Table[a[n],{n,1,84}] (* Jean-François Alcover, Mar 20 2011 *)

PARI

{ for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } \\ Harry J. Smith, Jul 10 2009

Python

from sympy import primerange as pr
def A060715(n): return len(list(pr(n+1, 2*n))) # Karl-Heinz Hofmann, May 05 2022

Formula

a(n) = Sum_{k=1..n-1} A010051(n+k). - Reinhard Zumkeller, Dec 03 2009

a(n) = pi(2n-1) - pi(n). - Wesley Ivan Hurt, Aug 21 2013

a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)). - Wesley Ivan Hurt, Jan 08 2022

Extensions

Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001

More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001