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A060715
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Number of primes between n and 2n exclusive.
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33
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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, 15, 15, 15, 15, 16, 15, 16
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| See the additional references and links mentioned in A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
a(A060756(n)) = n and a(m) <> n for m < A060756(n). [Reinhard Zumkeller, Jan 08 2012]
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REFERENCES
| M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
Math Olympiads, Bertrand's Postulate
R. Chapman, Bertrand postulate
S. Ramanujan, A Proof Of Bertrand's Postulate
M. Slone, PlanetMath.org, Proof of Bertrand's conjecture
Eric Weisstein's World of Mathematics, Bertrand's Postulate
Wolfram Research, Bertrand hypothesis
Wikipedia, Proof of Bertrand's postulate
C. K. Caldwell, The Prime Glossary, Bertrand's postulate
Dr. Wilkinson, The Math Forum, Erdos' Proof
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FORMULA
| a(n) = SUM(A010051(n+k): 1<=k<n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 03 2009]
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EXAMPLE
| a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.
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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:
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MATHEMATICA
| a[n_]:=PrimePi[2n-1]-PrimePi[n]; Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Mar 20 2011 *)
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PROG
| (PARI) { for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 10 2009]
(Haskell)
a060715 n = sum $ map a010051 [n+1..2*n-1] -- Reinhard Zumkeller, Jan 08 2012
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CROSSREFS
| Cf. A060756, A070046, A006992, A051501, A035250, A101909.
Cf. A000720, A014085, A104272, A143223, A143224, A143225, A143226, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
Sequence in context: A114920 A030361 * A108954 A123920 A029170 A079526
Adjacent sequences: A060712 A060713 A060714 * A060716 A060717 A060718
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KEYWORD
| nonn,easy
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 25 2001
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EXTENSIONS
| Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001. More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001
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