A217317 -id:A217317 - cp's OEIS frontend

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

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=', ')

A069922 Number of primes p such that n^n <= p <= n^n + n^2.

Original entry on oeis.org

1, 2, 2, 4, 1, 5, 4, 1, 2, 5, 1, 4, 4, 9, 7, 6, 2, 4, 7, 9, 7, 3, 7, 10, 10, 6, 12, 6, 10, 7, 8, 10, 7, 9, 13, 13, 7, 10, 11, 11, 9, 13, 11, 10, 15, 10, 11, 10, 19, 14, 16, 11, 16, 21, 20, 12, 9, 15, 21, 12, 10, 16, 15, 22, 19, 17, 18, 12, 19, 20, 13, 17, 13, 13, 17, 23

Offset: 1

Benoit Cloitre, May 05 2002

Question: for any n>0, is there at least one prime p such that n^n <= p <= n^n + n^2? In this case, that would be stronger than the Schinzel conjecture: "for m > 1 there's at least one prime p such that m <= p <= m + log(m)^2" since n^2 < log(n^n)^2 = n^2*log(n)^2.

Cf. A000040, A216266, A217317.

for(n=1,65,print1(sum(i=n^n,n^n+n^2,isprime(i)),","))

a(66)-a(76) from Alex Ratushnyak, Apr 20 2014

cp's OEIS Frontend

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

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