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A216265 Number of primes between n^3 - n and n^3.

%I A216265 #20 Mar 17 2013 17:26:31
%S A216265 0,1,0,1,0,1,1,1,1,2,2,2,0,2,3,2,2,2,2,1,2,3,4,1,3,3,2,3,3,3,2,1,3,2,
%T A216265 4,4,3,2,1,2,7,4,2,2,4,3,4,7,3,5,7,4,6,5,4,2,8,4,3,4,2,5,7,7,4,3,8,4,
%U A216265 1,3,2,10,4,5,4,6,7,8,6,6,1,6,8,8,7,7,6,7,4,10
%N A216265 Number of primes between n^3 - n and n^3.
%C A216265 Conjecture: a(n) > 0 for n > 13.
%H A216265 Alois P. Heinz, <a href="/A216265/b216265.txt">Table of n, a(n) for n = 1..10000</a>
%F A216265 a(n) = A000720(n^3) - A000720(n^3-n).
%e A216265 a(9) = 1 because between 9^3 - 9 and 9^3 there is just one prime (727).
%e A216265 a(10) = 2 because between 10^3 - 10 and 10^3 there are two primes (991 and 997).
%e A216265 a(11) = 2 because between 11^3 - 11 and 11^3 there are two primes (1321 and 1327).
%p A216265 a:= n-> add(`if`(isprime(t), 1, 0), t=n^3-n..n^3):
%p A216265 seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 17 2013
%t A216265 Table[PrimePi[n^3] - PrimePi[n^3 - n], {n, 100}] (* _Alonso del Arte_, Mar 17 2013 *)
%o A216265 (Java)
%o A216265 import java.math.BigInteger;
%o A216265 public class A216265 {
%o A216265     public static void main (String[] args) {
%o A216265       for (long n = 1; n < (1 << 21); n++) {
%o A216265         long cube = n*n*n, c = 0;
%o A216265         for (long k = cube - n; k < cube; ++k) {
%o A216265           BigInteger b1 = BigInteger.valueOf(k);
%o A216265           if (b1.isProbablePrime(2)) {
%o A216265             if (b1.isProbablePrime(80))
%o A216265               ++c;
%o A216265           }
%o A216265         }
%o A216265         System.out.printf("%d, ", c);
%o A216265       }
%o A216265     }
%o A216265 } // Ratushnyak
%o A216265 (PARI)
%o A216265 default(primelimit,10^7);
%o A216265 a(n) = primepi(n^3) - primepi(n^3-n);
%o A216265 /* _Joerg Arndt_, Mar 16 2013 */
%Y A216265 Cf. A094189, A216266.
%K A216265 nonn
%O A216265 1,10
%A A216265 _Alex Ratushnyak_, Mar 15 2013