A143226 Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2.

42, 55, 56, 58, 69, 77, 80, 119, 136, 137, 143, 145, 149, 156, 174, 177, 178, 188, 219, 225, 232, 247, 253, 254, 257, 261, 263, 297, 306, 310, 325, 327, 331, 335, 339, 341, 344, 356, 379, 395, 402, 410, 418, 421, 425, 433, 451, 485, 500

Offset

1,1

Comments

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.

It appears that this sequence is finite; searching up to 10^5, the last n appears to be 48717. [T. D. Noe, Aug 01 2008]

If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true for sufficiently large n. - Jonathan Sondow, Aug 02 2008

No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. - T. D. Noe, Aug 04 2008

See the additional reference and link to Ramanujan's work mentioned in A143223. - Jonathan Sondow, Aug 03 2008

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.

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.

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

A143223(n) < 0.

Example

There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 is a member.

Mathematica

L={}; Do[If[PrimePi[(n+1)^2]-PrimePi[n^2] < PrimePi[2n]-PrimePi[n], L=Append[L, n]], {n, 0, 500}]; L

Crossrefs

See A000720, A014085, A060715, A143223, A143224, A143225, A104272, A143227.

Sequence in context: A182147 A029695 A307986 * A043136 A039313 A043916

Adjacent sequences: A143223 A143224 A143225 * A143227 A143228 A143229

Keyword

nonn

Author

Jonathan Sondow, Jul 31 2008

Status

approved