OFFSET
1,2
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]
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
EXAMPLE
There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jul 31 2008
STATUS
approved