A117490 -id:A117490 - cp's OEIS frontend

A038107 Number of primes < n^2.

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

Offset: 0

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

nonn

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
a(n) - A000720(n) = A073882(n) - A010051(n) = A117490(n). - Reinhard Zumkeller, May 20 2010
A061265(a(n)) = 1 for n > 1. - Reinhard Zumkeller, Apr 15 2013
From Zhi-Wei Sun, Feb 17 2014: (Start)
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

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

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.)

T. D. Noe, Table of n, a(n) for n = 0..1000
Cino Hilliard, Sum of Primes.
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Wikipedia, Legendre's conjecture.

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

a038107 0 = 0
a038107 n = a000720 $ a000290 n
-- Reinhard Zumkeller, Apr 15 2013, Nov 01 2011

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

Table[PrimePi[n^2], {n, 0, 100}] (* Ray Chandler, Oct 22 2005 *)

a(n)=primepi(n^2) \\ Charles R Greathouse IV, Apr 26 2012

[prime_pi(n^2) for n in range(0, 59)] # Zerinvary Lajos, Jun 06 2009

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

a(n) ~ 1/2 * n^2/log n. - Charles R Greathouse IV, Apr 26 2012

Extended by Ray Chandler, Oct 22 2005

A073882 Number of primes between n and n^2.

Original entry on oeis.org

0, 2, 3, 4, 7, 8, 12, 14, 18, 21, 26, 29, 34, 38, 42, 48, 55, 59, 65, 70, 77, 84, 91, 96, 105, 113, 120, 128, 137, 144, 152, 161, 170, 180, 189, 199, 208, 216, 228, 239, 251, 261, 270, 281, 292, 305, 315, 327, 342, 352, 363, 378, 394, 405, 418, 429, 441, 458, 471

Offset: 1

Amarnath Murthy, Aug 16 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000720, A010051, A038107, A117490.

Table[c=PrimePi[n^2]-PrimePi[n];If[PrimeQ[n],c+1,c],{n,59}] (* James C. McMahon, Jan 12 2025 *)

PARI
```
a(n)=sum(k=n,n^2,isprime(k))
```

a(n) = A038107(n) - A000720(n) + A010051(n). - Reinhard Zumkeller, May 20 2010

a(n) = Sum_{k=n..n^2} A010051(k) = A117490(n) + A010051(n). - Reinhard Zumkeller, May 20 2010

Corrected and extended by Benoit Cloitre, Aug 20 2002

A117491 The number of primes between n and n^3 (with n and n^3 excluded).

Original entry on oeis.org

0, 3, 7, 16, 27, 44, 64, 93, 125, 164, 212, 264, 321, 394, 470, 558, 649, 758, 874, 999, 1139, 1290, 1448, 1624, 1812, 2011, 2218, 2451, 2697, 2951, 3217, 3501, 3806, 4126, 4472, 4810, 5182, 5567, 5983, 6401, 6837, 7295, 7775, 8279, 8796, 9347, 9901, 10484

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 22 2006

			For n = 3: between 3+1 = 4 and 3^3-1 = 26 there are the following primes: 5, 7, 11, 13, 17, 19, 23. Therefore the number of primes is 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A060715, A117490.

P:=proc(n) local i,j,np; for i from 1 by 1 to n do np:=0; for j from i+1 by 1 to i^3-1 do if isprime(j) then np:=np+1; fi; od; print(np); od; end: P(100);

Table[PrimePi[n^3]-PrimePi[n],{n,50}] (* Harvey P. Dale, Nov 08 2012 *)

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A038107 Number of primes < n^2.

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A073882 Number of primes between n and n^2.

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A117491 The number of primes between n and n^3 (with n and n^3 excluded).

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A093874 Least k such that there are at least n primes between k and k^2.

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