A143227 (Number of primes between n and 2n) - (number of primes between n^2 and (n+1)^2), if > 0.

1, 2, 1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 2, 6, 3, 3, 1, 1, 1, 2, 1, 1, 1, 1, 6, 3, 8, 3, 2, 3, 2, 3, 1, 1, 4, 3, 10, 2, 1, 1, 2, 3, 1, 3, 4, 2, 2, 9, 7, 2, 2, 4, 3, 3, 1, 2, 3, 5, 1, 2, 3, 2, 11, 3, 1, 2, 4, 7, 1, 1, 1, 1, 1, 5, 1, 2, 3, 3, 4, 2, 2, 9, 5, 1, 4, 2, 2

Offset

1,2

Comments

If the sequence is bounded (e.g., if it is finite), then Legendre's conjecture is true: there is always a prime between n^2 and (n+1)^2, at least for all sufficiently large n. This follows from the strong form of Bertrand's postulate proved by Ramanujan (see A104272 Ramanujan primes).

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.

Links

T. D. Noe, Table of n, a(n) for n=1..413

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.

T. D. Noe, Plot of the points (A143226(n), A143227(n))

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

E. W. Weisstein, Legendre's Conjecture in MathWorld

Formula

a(n) = |A143223(A143226(n))|.

Example

The first positive value of ((pi(2n) - pi(n)) - (pi((n+1)^2) - pi(n^2))) is 1 (at n = 42), the 2nd is 2 (at n = 55) and the 3rd is 1 (at n = 56), so a(1) = 1, a(2) = 2, a(3) = 1.

Mathematica

L={}; Do[ With[ {d=(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2])}, If[d>0, L=Append[L, d]]], {n, 0, 1000}]; L
Select[Table[(PrimePi[2n]-PrimePi[n])-(PrimePi[(n+1)^2]-PrimePi[n^2]), {n, 1000}], #>0&] (* Harvey P. Dale, Jun 19 2019 *)

Crossrefs

Cf. A000720, A014085, A060715, A104272, A143223, A143224, A143225, A143226 = corresponding values of n.

Sequence in context: A064823 A140225 A104758 * A329746 A302247 A026791

Adjacent sequences: A143224 A143225 A143226 * A143228 A143229 A143230

Keyword

nonn

Author

Jonathan Sondow, Aug 02 2008

Status

approved