OFFSET

0,3

COMMENTS

There is always a prime between n^2 and n^3 for n > 1. For n = 2, primes 5 and 7 are between 4 and 8. For n > 2, we have the number of primes between n^2 and n^3 ~ n^3/log(n^3) - n^2/log(n^2) = n^2*(2n-3)/(6*log(n)) -> infinity as n -> infinity. A corollary to this is that the number of primes is infinite.

Number of primes in row n of the triangle in A214084;

a(n) = Sum_{m=n^2..n^3} A010051(m). - Reinhard Zumkeller, Jul 07 2012

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..110

EXAMPLE

For n = 4 4^2 = 16, 4^3 = 64. there are 12 primes between 16 and 64 namely, 17,19,23,29,31,37,41,43,47,53,59,61.

PROG

(PARI) /* Count primes between x^2 and x^3. */ primex2x3(m, n) = { local(x, y, c); for(x=m, n, c=0; for(y=x^2, x^3, if(ispseudoprime(y), c++) ); print(c) ) }

(Haskell)

a079648 = sum . map a010051 . a214084_row -- Reinhard Zumkeller, Jul 07 2012

CROSSREFS

KEYWORD

nonn

AUTHOR

Cino Hilliard, Jan 22 2003, Aug 23 2007

EXTENSIONS

Edited by N. J. A. Sloane, Aug 22 2009 at the suggestion of Richard Stanley

STATUS

approved