A143224 - OEIS

A143224

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

0, 9, 36, 37, 46, 49, 85, 102, 107, 118, 122, 127, 129, 140, 157, 184, 194, 216, 228, 360, 365, 377, 378, 406, 416, 487, 511, 571, 609, 614, 672, 733, 767, 806, 813, 863, 869, 916, 923, 950, 978, 988, 1249, 1279, 1280, 1385, 1427, 1437, 1483, 1539, 1551, 1690

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OFFSET

1,2

COMMENTS

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

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.

REFERENCES

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

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

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]

LINKS

T. D. Noe, Table of n, a(n) for n=1..97 (no other n < 10^6)

T. Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate, arXiv:0807.3690 [math.GM], 2008.

M. Hassani, Counting primes in the interval (n^2,(n+1)^2), arXiv:math/0607096 [math.NT], 2006.

J. Pintz, Landau's problems on primes

S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.

J. Sondow, Ramanujan Prime in MathWorld.

J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld.

Eric Weisstein's World of Mathematics, Legendre's Conjecture.

FORMULA

A143223(a(n)) = 0.

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

(* Second program *)

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 *)

PROG

(PARI) is(n) = primepi((n+1)^2)-primepi(n^2)==primepi(2*n)-primepi(n) \\ Felix Fröhlich, Jul 25 2017

CROSSREFS

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

Cf. A104272, A143227. [Jonathan Sondow, Aug 03 2008]

Sequence in context: A020297 A330907 A231666 * A192610 A319958 A329808

Adjacent sequences: A143221 A143222 A143223 * A143225 A143226 A143227

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jul 31 2008

STATUS

approved