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A217317
Number of primes between n^2 and n^2 + log_2(n)^2 (inclusive).
3
0, 1, 1, 2, 1, 2, 1, 3, 2, 4, 2, 2, 3, 2, 4, 4, 1, 2, 3, 2, 3, 4, 2, 3, 3, 3, 4, 2, 4, 3, 4, 4, 5, 3, 4, 6, 2, 5, 3, 7, 4, 4, 5, 2, 4, 5, 4, 3, 3, 3, 4, 6, 3, 3, 3, 4, 5, 4, 3, 5, 3, 5, 3, 4, 7, 4, 6, 6, 4, 6, 3, 3, 3, 6, 7, 6, 2, 5, 6, 2, 6, 4, 4, 3, 5, 3, 7
OFFSET
1,4
COMMENTS
Indices of zeros: 1, 1165, 4292936, 4765516.
Conjecture: a(n) > 0 for n > 4765516.
Conjecture checked up to n = 5 * 10^10. - Charles R Greathouse IV, Mar 21 2013
Conjecture checked up to 4 * 10^18. Note that this conjecture is consistent with Granville's conjecture that lim sup (prime(n+1)-prime(n))/log(prime(n))^2 >= 2/e^gamma, where gamma is Euler's constant. - Charles R Greathouse IV, Mar 21 2016
MATHEMATICA
Table[Length[Select[Range[n^2, n^2 + Log[2, n]^2], PrimeQ]], {n, 100}] (* T. D. Noe, Mar 21 2013 *)
Table[PrimePi[n^2+Log[2, n]^2]-PrimePi[n^2], {n, 90}] (* Harvey P. Dale, May 22 2014 *)
PROG
(Python)
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 + int(math.log(n, 2)**2) + 1
for i in range(n*n+1, top):
if i&1: c += isprime(i)
print(str(c), end=', ')
(PARI) a(n)=sum(i=n^2+1, n^2+(log(n)/log(2))^2, isprime(i)) \\ Charles R Greathouse IV, Mar 21 2013
CROSSREFS
Sequence in context: A194868 A304574 A139024 * A331128 A154958 A025806
KEYWORD
nonn
AUTHOR
Alex Ratushnyak, Mar 20 2013
STATUS
approved