|
%I #43 Sep 08 2022 08:44:52
%S 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,
%T 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,
%U 13,13,12,12,13,13,14,14,13,14,15,15,14,14,13,14,15
%N Number of primes between n and 2n (inclusive).
%C 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.
%C The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - _Lekraj Beedassy_, Jan 01 2007
%C The number of partitions of 2n into exactly two parts with first part prime, n > 1. - _Wesley Ivan Hurt_, Jun 15 2013
%D Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.
%H T. D. Noe, <a href="/A035250/b035250.txt">Table of n, a(n) for n = 1..1000</a>
%H International Mathematics Olympiad, <a href="https://web.archive.org/web/20170407203826/http://www.mathsolympiad.org.nz/wp-content/uploads/2009/01/bertrands-theorem.pdf">Proof of Bertrand's Postulate</a> [Via Wayback Machine]
%F a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - _Reinhard Zumkeller_, Jul 05 2010
%F a(A059316(n)) = n and a(m) <> n for m < A059316(n). - _Reinhard Zumkeller_, Jan 08 2012
%F a(n) = sum(A010051(k): k=n..2*n). [_Reinhard Zumkeller_, Jan 08 2012]
%F a(n) = pi(2n) - pi(n-1). [_Wesley Ivan Hurt_, Jun 15 2013]
%e The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
%e 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.
%p with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # _Wesley Ivan Hurt_, Aug 09 2014
%t f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* _Robert G. Wilson v_, Dec 23 2012 *)
%o (Haskell)
%o a035250 n = sum $ map a010051 [n..2*n] -- _Reinhard Zumkeller_, Jan 08 2012
%o (Magma) [#PrimesInInterval(n, 2*n): n in [1..80]]; // _Bruno Berselli_, Sep 05 2012
%o (PARI) a(n)=primepi(2*n)-primepi(n-1) \\ _Charles R Greathouse IV_, Jul 01 2013
%Y Cf. A073837, A073838, A099802, A060715.
%K nonn
%O 1,2
%A _Erich Friedman_
|
