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%I #90 Apr 03 2023 10:36:10
%S 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,
%T 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,
%U 12,12,12,13,13,14,13,13,14,15,14,14,13,13,14,15,15
%N Number of primes between n and 2n exclusive.
%C See the additional references and links mentioned in A143227. - _Jonathan Sondow_, Aug 03 2008
%C a(A060756(n)) = n and a(m) <> n for m < A060756(n). - _Reinhard Zumkeller_, Jan 08 2012
%C For prime n conjecturally a(n) = A226859(n). - _Vladimir Shevelev_, Jun 27 2013
%C 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
%D M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.
%H T. D. Noe, <a href="/A060715/b060715.txt">Table of n, a(n) for n = 1..1000</a>
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/Bertrandspostulate.html">Bertrand's postulate</a>
%H R. Chapman, <a href="http://www.maths.ex.ac.uk/~rjc/etc/bertrand.pdf">Bertrand postulate</a> [Broken link]
%H Math Olympiads, <a href="http://matholymp.com/TUTORIALS/Bertrand.pdf">Bertrand's Postulate</a> [Broken link]
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, arXiv:0909.0715v13 [math.NT]
%H Vladimir Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
%H M. Slone, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/ProofOfBertrandsConjecture.html">Proof of Bertrand's conjecture</a>
%H Jonathan Sondow and Eric Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate</a>, World of Mathematics
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate">Proof of Bertrand's postulate</a>
%H Dr. Wilkinson, The Math Forum, <a href="http://mathforum.org/library/drmath/view/51527.html">Erdos' Proof</a>
%H Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/Prime/31/03/ShowAll.html">Bertrand hypothesis</a>
%F a(n) = Sum_{k=1..n-1} A010051(n+k). - _Reinhard Zumkeller_, Dec 03 2009
%F a(n) = pi(2n-1) - pi(n). - _Wesley Ivan Hurt_, Aug 21 2013
%F a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)). - _Wesley Ivan Hurt_, Jan 08 2022
%e a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.
%p 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:
%p with(numtheory); seq(pi(2*k-1)-pi(k),k=1..100); # _Wesley Ivan Hurt_, Aug 21 2013
%t a[n_]:=PrimePi[2n-1]-PrimePi[n]; Table[a[n],{n,1,84}] (* _Jean-François Alcover_, Mar 20 2011 *)
%o (PARI) { for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } \\ _Harry J. Smith_, Jul 10 2009
%o (Haskell)
%o a060715 n = sum $ map a010051 [n+1..2*n-1] -- _Reinhard Zumkeller_, Jan 08 2012
%o (Magma) [0] cat [#PrimesInInterval(n+1, 2*n-1): n in [2..80]]; // _Bruno Berselli_, Sep 05 2012
%o (Python) from sympy import primerange as pr
%o def A060715(n): return len(list(pr(n+1, 2*n))) # _Karl-Heinz Hofmann_, May 05 2022
%Y Cf. A060756, A070046, A006992, A051501, A035250, A101909.
%Y Cf. A000720, A014085, A104272, A143227, A128076.
%Y Cf. A143223, A143224, A143225, A143226.
%K nonn,easy
%O 1,4
%A _Lekraj Beedassy_, Apr 25 2001
%E Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001
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