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A038107 Number of primes < n^2. 41

%I #58 May 16 2021 05:54:20

%S 0,0,2,4,6,9,11,15,18,22,25,30,34,39,44,48,54,61,66,72,78,85,92,99,

%T 105,114,122,129,137,146,154,162,172,181,191,200,210,219,228,240,251,

%U 263,274,283,295,306,319,329,342,357,367,378,393,409,421,434,445,457,474

%N Number of primes < n^2.

%C Also number of primes <= n^2 since n^2 is not prime.

%C Also the number of primes contained within an n X n square spiral. - _William A. Tedeschi_, Mar 03 2008

%C 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

%C a(n) - A000720(n) = A073882(n) - A010051(n) = A117490(n). - _Reinhard Zumkeller_, May 20 2010

%C A061265(a(n)) = 1 for n > 1. - _Reinhard Zumkeller_, Apr 15 2013

%C From _Zhi-Wei Sun_, Feb 17 2014: (Start)

%C Conjecture:

%C (i) The sequence a(n)^(1/n) (n = 3, 4, ...) is strictly decreasing (to the limit 1).

%C (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)

%C A classical conjecture of Legendre asserts that a(n) < a(n+1) for all n > 0.

%C 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

%D 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.)

%H T. D. Noe, <a href="/A038107/b038107.txt">Table of n, a(n) for n = 0..1000</a>

%H Cino Hilliard, <a href="http://docs.google.com/Doc?id=dgpq9w4b_26dtrq634m">Sum of Primes</a>.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Legendre&#39;s_conjecture">Legendre's conjecture</a>.

%F a(n) = A000720(A000290(n)).

%F a(n) ~ 1/2 * n^2/log n. - _Charles R Greathouse IV_, Apr 26 2012

%e a(2)=2 because the only primes < 4 are 2 and 3.

%p A038107 := proc(n) numtheory[pi]( n^2) ; end: seq(A038107(n),n=0..100) ; # _R. J. Mathar_, Jun 22 2009

%t Table[PrimePi[n^2], {n, 0, 100}] (* _Ray Chandler_, Oct 22 2005 *)

%o (Sage) [prime_pi(n^2) for n in range(0, 59)] # _Zerinvary Lajos_, Jun 06 2009

%o (Haskell)

%o a038107 0 = 0

%o a038107 n = a000720 $ a000290 n

%o -- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011

%o (PARI) a(n)=primepi(n^2) \\ _Charles R Greathouse IV_, Apr 26 2012

%Y Cf. A014085 (first differences), A111208, A194189, A262408, A262443, A262447, A262462.

%K nonn

%O 0,3

%A Joe K. Crump (joecr(AT)carolina.rr.com)

%E Extended by _Ray Chandler_, Oct 22 2005

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)