A079648 - OEIS

%I #15 Jun 22 2019 10:13:23

%S 0,0,2,5,12,21,36,53,79,107,143,187,235,288,356,428,510,595,699,810,

%T 929,1062,1206,1358,1528,1707,1898,2098,2323,2561,2807,3066,3340,3636,

%U 3946,4283,4611,4975,5351,5755,6162,6587,7034,7506,7998,8504,9042,9587,10157

%N Number of primes between n^2 and n^3.

%C 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.

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

%C a(n) = Sum_{m=n^2..n^3} A010051(m). - _Reinhard Zumkeller_, Jul 07 2012

%H Reinhard Zumkeller, <a href="/A079648/b079648.txt">Table of n, a(n) for n = 0..110</a>

%e 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.

%o (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) ) }

%o (Haskell)

%o a079648 = sum . map a010051 . a214084_row  -- _Reinhard Zumkeller_, Jul 07 2012

%K nonn

%O 0,3

%A _Cino Hilliard_, Jan 22 2003, Aug 23 2007

%E Edited by _N. J. A. Sloane_, Aug 22 2009 at the suggestion of _Richard Stanley_