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

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, 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, 1, 3, 2, 10, 4, 5, 4, 6, 7, 8, 6, 6, 1, 6, 8, 8, 7, 7, 6, 7, 4, 10

Offset: 1

Author

Alex Ratushnyak, Mar 15 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for n > 13.

Examples

			a(9) = 1 because between 9^3 - 9 and 9^3 there is just one prime (727).
a(10) = 2 because between 10^3 - 10 and 10^3 there are two primes (991 and 997).
a(11) = 2 because between 11^3 - 11 and 11^3 there are two primes (1321 and 1327).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A094189, A216266.

Programs

Java

import java.math.BigInteger;
public class A216265 {
    public static void main (String[] args) {
      for (long n = 1; n < (1 << 21); n++) {
        long cube = n*n*n, c = 0;
        for (long k = cube - n; k < cube; ++k) {
          BigInteger b1 = BigInteger.valueOf(k);
          if (b1.isProbablePrime(2)) {
            if (b1.isProbablePrime(80))
              ++c;
          }
        }
        System.out.printf("%d, ", c);
      }
    }
} // Ratushnyak

Maple

a:= n-> add(`if`(isprime(t), 1, 0), t=n^3-n..n^3):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2013

Mathematica

Table[PrimePi[n^3] - PrimePi[n^3 - n], {n, 100}] (* Alonso del Arte, Mar 17 2013 *)

PARI

default(primelimit,10^7);
a(n) = primepi(n^3) - primepi(n^3-n);
/* Joerg Arndt, Mar 16 2013 */

Formula

a(n) = A000720(n^3) - A000720(n^3-n).