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A038107
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Number of primes < n^2.
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42
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0, 0, 2, 4, 6, 9, 11, 15, 18, 22, 25, 30, 34, 39, 44, 48, 54, 61, 66, 72, 78, 85, 92, 99, 105, 114, 122, 129, 137, 146, 154, 162, 172, 181, 191, 200, 210, 219, 228, 240, 251, 263, 274, 283, 295, 306, 319, 329, 342, 357, 367, 378, 393, 409, 421, 434, 445, 457, 474
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OFFSET
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0,3
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COMMENTS
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Also number of primes <= n^2 since n^2 is not prime.
Also the number of primes contained within an n X n square spiral. - William A. Tedeschi, Mar 03 2008
For large n, these numbers closely approximate the sum of primes less than n. For example, n = 10^10, sum of primes < n = 2220822432581729238. The number of primes < (10^10)^2 = 10^20 = 2220819602560918840. The error is 0.0000012743... The derivation of this is in the link Sum of Primes. - Cino Hilliard, Jun 09 2008
Conjecture:
(i) The sequence a(n)^(1/n) (n = 3, 4, ...) is strictly decreasing (to the limit 1).
(ii) If n > 0 is not among 25, 35, 44, 46, 105, then the interval [a(n), a(n+1)] contains at least one prime. (End)
A classical conjecture of Legendre asserts that a(n) < a(n+1) for all n > 0.
Conjecture: All the numbers Sum_{i=j,...,k} 1/a(i) with 1 < j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
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REFERENCES
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Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187. (See Conjectures 2.14-2.16.)
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LINKS
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FORMULA
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EXAMPLE
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a(2)=2 because the only primes < 4 are 2 and 3.
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MAPLE
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MATHEMATICA
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Table[PrimePi[n^2], {n, 0, 100}] (* Ray Chandler, Oct 22 2005 *)
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PROG
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(Sage) [prime_pi(n^2) for n in range(0, 59)] # Zerinvary Lajos, Jun 06 2009
(Haskell)
a038107 0 = 0
a038107 n = a000720 $ a000290 n
-- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Joe K. Crump (joecr(AT)carolina.rr.com)
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EXTENSIONS
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STATUS
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approved
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