A143225 Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.

0, 3, 9, 9, 10, 10, 16, 20, 19, 21, 23, 23, 24, 25, 28, 31, 32, 36, 38, 56, 57, 59, 59, 62, 65, 71, 75, 84, 88, 88, 96, 102, 107, 115, 116, 119, 120, 126, 125, 129, 132, 132, 163, 168, 168, 182, 189, 189, 192, 197, 198, 213, 236

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.

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

Examples

			There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 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..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
• E. W. Weisstein, Legendre's Conjecture in MathWorld

Crossrefs

See A000720, A014085, A060715, A143223, A143224, A143226.

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

Programs

Mathematica

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

Formula

a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0.