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A060715 Number of primes between n and 2n exclusive. 44
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

REFERENCES

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

LINKS

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

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

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

EXAMPLE

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

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-Fran├žois Alcover, Mar 20 2011 *)

PROG

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

(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

(Python) from sympy import primerange as pr

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

CROSSREFS

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

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

Cf. A143223, A143224, A143225, A143226.

Sequence in context: A114920 A283190 A030361 * A108954 A123920 A322141

Adjacent sequences:  A060712 A060713 A060714 * A060716 A060717 A060718

KEYWORD

nonn,easy

AUTHOR

Lekraj Beedassy, Apr 25 2001

EXTENSIONS

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

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

STATUS

approved

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Last modified September 26 22:02 EDT 2022. Contains 357051 sequences. (Running on oeis4.)