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

Author

Jonathan Sondow, Jul 31 2008

Keywords

• nonn

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

Examples

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

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

Crossrefs

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

Programs

Mathematica

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

Formula

A143223(n) < 0.