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A143225
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Number of primes between n^2 and (n+1)^2, if equal to the number of primes between n and 2n.
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9
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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
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OFFSET
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1,2
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COMMENTS
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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. [From Jonathan Sondow, Aug 03 2008]
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REFERENCES
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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, "A Proof of Bertrand's Postulate," J. Indian Math. Soc. 11 (1919) 181-182.
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LINKS
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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
M. Hassani, Counting primes in the interval (n^2,(n+1)^2)
J. Pintz, Landau's problems on primes
J. Sondow, Ramanujan Prime in MathWorld [From Jonathan Sondow, Aug 02 2008]
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow, Aug 02 2008]
E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow, Aug 02 2008]
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FORMULA
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a(n) = A014085(A143224(n)) = A060715(A143224(n)) for n > 0
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EXAMPLE
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There are 3 primes between 9^2 and 10^2 and 3 primes between 9 and 2*9, so 3 is a member.
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MATHEMATICA
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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
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CROSSREFS
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See A000720, A014085, A060715, A143223, A143224, A143226.
Cf. A104272, A143227. [From Jonathan Sondow, Aug 03 2008]
Sequence in context: A004166 A110759 A063750 * A223195 A203600 A099720
Adjacent sequences: A143222 A143223 A143224 * A143226 A143227 A143228
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KEYWORD
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nonn
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AUTHOR
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Jonathan Sondow, Jul 31 2008
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STATUS
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approved
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