A214084 -id:A214084 - cp's OEIS frontend

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

Cino Hilliard, Jan 22 2003, Aug 23 2007

nonn

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

			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.

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

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

/* 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) ) }

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

A214085 n^2 * (n^4 - n^2 + n + 1) / 2.

Original entry on oeis.org

0, 1, 30, 342, 1960, 7575, 22806, 57820, 129312, 262845, 495550, 879186, 1483560, 2400307, 3747030, 5671800, 8358016, 12029625, 16956702, 23461390, 31924200, 42790671, 56578390, 73884372, 95392800, 121883125, 154238526, 193454730, 240649192, 297070635

Offset: 0

Reinhard Zumkeller, Jul 07 2012

Row sums of the triangle in A214084.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

a214085 n = n^2 * (n^4 - n^2 + n + 1) `div` 2

[n^2*(n^4-n^2+n+1)/2: n in [0..29]]; // Bruno Berselli, Jul 09 2012

Table[n^2 (n^4 - n^2 + n + 1)/2, {n, 0, 29}] (* Bruno Berselli, Jul 09 2012 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,30,342,1960,7575,22806},40] (* Harvey P. Dale, Dec 12 2012 *)

a(n) = n * A000217(n) * A100104(n).
a(n) = A000217(A000578(n)) - A000217(A000290(n) - 1).
G.f.: x*(1+23*x+153*x^2+161*x^3+22*x^4)/(1-x)^7. - Bruno Berselli, Jul 09 2012
a(0)=0, a(1)=1, a(2)=30, a(3)=342, a(4)=1960, a(5)=7575, a(6)=22806, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Dec 12 2012

cp's OEIS Frontend

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

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