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A217317
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Number of primes between n^2 and n^2 + log_2(n)^2 (inclusive).
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3
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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
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OFFSET
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1,4
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COMMENTS
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Indices of zeros: 1, 1165, 4292936, 4765516.
Conjecture: a(n) > 0 for n > 4765516.
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
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(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)+', ',
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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