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Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).
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%I #26 Feb 23 2023 04:21:05

%S 0,9,36,37,46,49,85,102,107,118,122,127,129,140,157,184,194,216,228,

%T 360,365,377,378,406,416,487,511,571,609,614,672,733,767,806,813,863,

%U 869,916,923,950,978,988,1249,1279,1280,1385,1427,1437,1483,1539,1551,1690

%N Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n).

%C The sequence gives the positions of zeros in A143223. The number of primes in question is A143225(n).

%C 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.

%D M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer, NY, 2001.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1989, p. 19.

%D 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. [_Jonathan Sondow_, Aug 03 2008]

%H T. D. Noe, <a href="/A143224/b143224.txt">Table of n, a(n) for n=1..97</a> (no other n < 10^6)

%H 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 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 J. Pintz, <a href="http://www.renyi.hu/~pintz/pjapr.pdf">Landau's problems on primes</a>

%H 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 J. Sondow, <a href="http://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>.

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate in MathWorld</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendresConjecture.html">Legendre's Conjecture</a>.

%F A143223(a(n)) = 0.

%e There is the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 is a term.

%p with(numtheory): A143224:=n->`if`(pi((n+1)^2)-pi(n^2) = pi(2*n)-pi(n), n, NULL): seq(A143224(n), n=0..2000); # _Wesley Ivan Hurt_, Jul 25 2017

%t L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] == PrimePi[2n]-PrimePi[n], L=Append[L,n]], {n,0,2000}]; L

%t (* Second program *)

%t With[{nn = 2000}, {0}~Join~Position[#, {0}][[All, 1]] &@ Map[Differences, Transpose@ {Differences@ Array[PrimePi[#^2] &, nn], Array[PrimePi[2 #] - PrimePi[#] &, nn - 1]}]] (* _Michael De Vlieger_, Jul 25 2017 *)

%o (PARI) is(n) = primepi((n+1)^2)-primepi(n^2)==primepi(2*n)-primepi(n) \\ _Felix Fröhlich_, Jul 25 2017

%Y See A000720, A014085, A060715, A143223, A143225, A143226.

%Y Cf. A104272, A143227. [_Jonathan Sondow_, Aug 03 2008]

%K nonn

%O 1,2

%A _Jonathan Sondow_, Jul 31 2008