This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143223 #24 Feb 16 2025 08:33:08
%S A143223 0,2,1,1,1,1,2,1,2,0,1,1,1,2,1,2,2,1,2,2,3,2,1,1,3,2,1,1,2,2,1,3,2,3,
%T A143223 1,2,0,0,3,2,2,2,-1,3,2,3,0,4,6,0,1,4,4,1,1,-2,-1,3,-1,3,3,1,5,3,1,3,
%U A143223 1,2,4,-1,6,1,1,4,4,4,7,-1,3,8,-2,5,3,5,1,0,5,5,1,2,3,2,1,5,3,3,2,3,4,1,2
%N A143223 (Number of primes between n^2 and (n+1)^2) - (number of primes between n and 2n).
%C A143223 Legendre's conjecture (still open) says there is always a prime between n^2 and (n+1)^2. Bertrand's postulate (actually a theorem due to Chebyshev) says there is always a prime between n and 2n.
%C A143223 Hashimoto's plot of (1 - a(n)) shows that |a(n)| is small compared to n for n < 30000.
%C A143223 From _Jonathan Sondow_, Aug 07 2008: (Start)
%C A143223 It appears that there are only a finite number of negative terms (see A143226).
%C A143223 If the negative terms are bounded, then Legendre's conjecture is true, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes). (End)
%D A143223 M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.
%D A143223 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.
%D A143223 S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209.
%H A143223 Vincenzo Librandi, <a href="/A143223/b143223.txt">Table of n, a(n) for n = 0..10000</a>
%H A143223 T. Hashimoto, <a href="http://arxiv.org/abs/0807.3690">On a certain relation between Legendre's conjecture and Bertrand's postulate</a>, arXiv:0807.3690 [math.GM], 2008.
%H A143223 M. Hassani, <a href="http://arXiv.org/abs/math/0607096">Counting primes in the interval (n^2,(n+1)^2)</a>, arXiv:math/0607096 [math.NT], 2006.
%H A143223 T. D. Noe, <a href="http://www.sspectra.com/math/A143223.gif">Plot of A143223(n) for n to 10^6</a>
%H A143223 J. Pintz, <a href="http://www.renyi.hu/~pintz/">Landau's problems on primes</a>
%H A143223 S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H A143223 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate in MathWorld</a>
%H A143223 J. Sondow, <a href="https://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>
%H A143223 E. W. Weisstein, <a href="https://mathworld.wolfram.com/LegendresConjecture.html">Legendre's Conjecture in MathWorld</a>
%F A143223 a(n) = A014085(n) - A060715(n) (for n > 0) = [pi((n+1)^2) - pi(n^2)] - [pi(2n) - pi(n)] (for n > 1).
%e A143223 There are 4 primes between 6^2 and 7^2 and 2 primes between 6 and 2*6, so a(6) = 4 - 2 = 2.
%e A143223 a(1) = 2 because there are two primes between 1^2 and 2^2 (namely, 2 and 3) and none between 1 and 2. [_Jonathan Sondow_, Aug 07 2008]
%t A143223 L={0,2}; Do[L=Append[L,(PrimePi[(n+1)^2]-PrimePi[n^2]) - (PrimePi[2n]-PrimePi[n])], {n,2,100}]; L
%o A143223 (PARI) a(n)=sum(k=n^2+1,n^2+2*n,isprime(k))-sum(k=n+1,2*n,isprime(k)) \\ _Charles R Greathouse IV_, May 30 2014
%Y A143223 See A000720, A014085, A060715, A143224, A143225, A143226.
%Y A143223 Negative terms are A143227. Cf. A104272 (Ramanujan primes).
%K A143223 sign
%O A143223 0,2
%A A143223 _Jonathan Sondow_, Jul 31 2008
%E A143223 Corrected by _Jonathan Sondow_, Aug 07 2008, Aug 09 2008