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

0,3

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

Sequence in context: A258602 A327065 A307605 * A080838 A244396 A182993

Adjacent sequences: A079645 A079646 A079647 * A079649 A079650 A079651

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