|
| |
|
|
A035250
|
|
Number of primes between n and 2n (inclusive).
|
|
35
|
|
|
|
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, 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, 13, 13, 12, 12, 13, 13, 14, 14, 13, 14, 15, 15, 14, 14, 13, 14, 15
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
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.
The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy, Jan 01 2007
The number of partitions of 2n into exactly two parts with first part prime, n > 1. - Wesley Ivan Hurt, Jun 15 2013
|
|
|
REFERENCES
|
Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 1..1000
International Mathematics Olympiad, Proof of Bertrand's Postulate [Via Wayback Machine]
|
|
|
FORMULA
|
a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - Reinhard Zumkeller, Jul 05 2010
a(A059316(n)) = n and a(m) <> n for m < A059316(n). - Reinhard Zumkeller, Jan 08 2012
a(n) = sum(A010051(k): k=n..2*n). [Reinhard Zumkeller, Jan 08 2012]
a(n) = pi(2n) - pi(n-1). [Wesley Ivan Hurt, Jun 15 2013]
|
|
|
EXAMPLE
|
The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
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.
|
|
|
MAPLE
|
with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014
|
|
|
MATHEMATICA
|
f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* Robert G. Wilson v, Dec 23 2012 *)
|
|
|
PROG
|
(Haskell)
a035250 n = sum $ map a010051 [n..2*n] -- Reinhard Zumkeller, Jan 08 2012
(Magma) [#PrimesInInterval(n, 2*n): n in [1..80]]; // Bruno Berselli, Sep 05 2012
(PARI) a(n)=primepi(2*n)-primepi(n-1) \\ Charles R Greathouse IV, Jul 01 2013
|
|
|
CROSSREFS
|
Cf. A073837, A073838, A099802, A060715.
Sequence in context: A257212 A001031 A336543 * A165054 A067743 A029230
Adjacent sequences: A035247 A035248 A035249 * A035251 A035252 A035253
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Erich Friedman
|
|
|
STATUS
|
approved
|
| |
|
|
