

A050216


Number of primes between (prime(n))^2 and (prime(n+1))^2, with a(0) = 2 by convention.


15



2, 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, 57, 44, 20, 46, 80, 78, 32, 90, 66, 30, 106, 75, 114, 163, 89, 42, 87, 42, 100, 354, 99, 165, 49, 299, 58, 182, 186, 128, 198, 195, 76, 356, 77, 144, 75, 463, 479, 168, 82, 166, 270, 90, 438, 275, 274, 292, 91, 292, 199, 99
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OFFSET

0,1


COMMENTS

The function in Brocard's Conjecture, which states that for n >= 2, a(n) >= 4.
The lines in the graph correspond to prime gaps of 2, 4, 6, ... .  T. D. Noe, Feb 04 2008
Lengths of blocks of consecutive primes in A000430 (union of primes and squares of primes).  Reinhard Zumkeller, Sep 23 2011
In the nth step of the sieve of Eratosthenes, all multiples of prime(n) are removed. Then a(n) gives the number of new primes obtained after the nth step.  JeanChristophe HervĂ©, Oct 27 2013
More precisely, after the nth step, one is sure to have eliminated all composites less than prime(n+1)^2, since any composite N has a prime factor <= sqrt(N). It is in exactly this (restricted) sense that a(n) yields the number of "new primes" (additional numbers known to be prime) after the nth step. But one knows after the nth step also that all remaining numbers between prime(n+1)^2 and prime(n+1)*(prime(n+1)+2) are prime: By construction they don't have a factor less than prime(n+1) and they don't have a factor prime(n+1) so the least prime factor could be prime(n+2) >= prime(n+1)+2. For example, after eliminating multiples of 3 in the 2nd step, one has (2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 31, 35, ...) and one knows that all remaining numbers strictly in between 5^2=25 and 5*(5+2)=35 are prime, too.  M. F. Hasler, Dec 31 2014
Numerically, the slope of the lowest "ray" m(n) = min {a(k); k>n}, seems to converge to a value somewhere in the range 1.75 < m(n)/n < 1.8; with m(n)/n > 1.7 for n > 900, m(n)/n > 1.75 for n > 2700.  M. F. Hasler, Dec 31 2014
Legendre's conjecture (see A014085) would imply that a(n) >= 2 for all n and that sequences A054272, A250473 and A250474 were thus strictly increasing (see the Wikipedia article about Brocard's conjecture).  Antti Karttunen, Jan 01 2015


LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000
A. G. Shannon and J. V. Leyendekkers, "On Legendre's Conjecture", Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017): 117125.
Eric Weisstein's World of Mathematics, Brocard's Conjecture
Wikipedia, Brocard's Conjecture


FORMULA

For all n >= 1, a(n) = A256468(n) + A256469(n).  Antti Karttunen, Mar 30 2015


EXAMPLE

There are 2 primes less than 2^2, there are 2 primes between 2^2 and 3^2, 5 primes between 3^2 and 5^2, etc.


MATHEMATICA

PrimePi[ Prime[ n+1 ]^2 ]PrimePi[ Prime[ n ]^2 ]


PROG

(Haskell)
import Data.List (group)
a050216 n = a050216_list !! (n1)
a050216_list =
map length $ filter (/= [0]) $ group $ map a010051 a000430_list
 Reinhard Zumkeller, Sep 23 2011
(PARI) a(n)={nreturn(2); primepi(prime(n+1)^2)primepi(prime(n)^2)} \\ M. F. Hasler, Dec 31 2014


CROSSREFS

First differences of A000879.
One more than A251723.
Cf. A010051, A001248, A014085, A089609, A251719, A256468, A256469, A256470.
Sequence in context: A304867 A303931 A028410 * A320296 A295198 A080880
Adjacent sequences: A050213 A050214 A050215 * A050217 A050218 A050219


KEYWORD

nonn,look


AUTHOR

Eric W. Weisstein


EXTENSIONS

Edited by N. J. A. Sloane, Nov 15 2009
Example corrected by Jonathan Sperry, Aug 30 2013


STATUS

approved



