A216265 -id:A216265 - cp's OEIS frontend

A216266 Number of primes between n^3 and n^3+n (inclusive).

1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 2, 1, 2, 1, 3, 3, 3, 2, 4, 0, 3, 5, 4, 4, 2, 3, 2, 2, 5, 3, 3, 2, 5, 2, 3, 4, 5, 2, 3, 3, 5, 8, 5, 4, 5, 4, 3, 6, 6, 4, 4, 6, 5, 3, 7, 8, 2, 3, 6, 6, 5, 4, 5, 6, 5, 4, 4, 3, 4, 8, 8, 4, 5, 8, 7, 6, 5, 4, 5, 9, 6, 8, 8, 6, 8, 10, 6, 9, 11

Offset: 1

Alex Ratushnyak, Mar 15 2013

nonn

Conjecture: a(n)>0 for n>23.

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

Cf. A089610, A216265.

import java.math.BigInteger;
public class A216266 {
    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+1; k<=cube+n; ++k) {
          BigInteger b1 = BigInteger.valueOf(k);
          if (b1.isProbablePrime(2)) {
            if (b1.isProbablePrime(80))
              ++c;
          }
        }
        System.out.printf("%d, ", c);
      }
    }
}

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

Table[PrimePi[n^3+n]-PrimePi[n^3],{n,100}] (* Harvey P. Dale, Apr 19 2014 *)

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

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

A217721 Number of primes between n^2 - log_2(n)^2 and n^2 (inclusive).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 3, 3, 3, 5, 4, 4, 4, 2, 2, 4, 5, 2, 5, 2, 3, 4, 4, 3, 4, 5, 5, 3, 5, 7, 2, 3, 6, 6, 4, 5, 3, 3, 5, 6, 4, 5, 3, 3, 4, 4, 4, 4, 4, 4, 3, 5, 5, 4, 4, 2, 4, 4, 5, 5, 6, 5, 6, 5, 4, 6, 2, 7

Offset: 1

Alex Ratushnyak, Mar 21 2013

nonn

Conjecture: a(n) > 0 for n > 1.

Conjecture checked up to n = 2^28 - 1.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A094189, A216265, A217317.

Table[Length[Select[Range[n^2, n^2 - Log[2, n]^2, -1], PrimeQ]], {n, 100}] (* T. D. Noe, Mar 21 2013 *)

import math
def isprime(k):
  s = 3
  while s*s <= k:
    if k%s==0:  return 0
    s+=2
  return 1
for n in range(1, 333):
  c = 0
  top = n*n
  for i in range(top - int(math.log(n, 2)**2), top):
    if i&1:  c += isprime(i)
  print(str(c), end=', ')

cp's OEIS Frontend

A216266 Number of primes between n^3 and n^3+n (inclusive).

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