A079648 Number of primes between n^2 and n^3.

0, 0, 2, 5, 12, 21, 36, 53, 79, 107, 143, 187, 235, 288, 356, 428, 510, 595, 699, 810, 929, 1062, 1206, 1358, 1528, 1707, 1898, 2098, 2323, 2561, 2807, 3066, 3340, 3636, 3946, 4283, 4611, 4975, 5351, 5755, 6162, 6587, 7034, 7506, 7998, 8504, 9042, 9587, 10157

Offset: 0

Author

Cino Hilliard, Jan 22 2003, Aug 23 2007

Keywords

• nonn

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

Examples

			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.
		

Links

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

Programs

Haskell

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

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) ) }

Extensions

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