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Search: id:A143226
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| A143226 |
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Numbers n such that there are more primes between n and 2n than between n^2 and (n+1)^2. |
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10
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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 Chebychev) 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. [From T. D. Noe (noe(AT)sspectra.com), Aug 01 2008]
If the sequence is finite, then, by Bertrand's postulate, Legendre's conjecture is true, at least for all sufficiently large n. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
No other n <= 10^6. The plot of A143223 shows that it is quite likely that there are no additional terms. [From T. D. Noe (noe(AT)sspectra.com), Aug 04 2008]
See the additional reference and link to Ramanujan's work mentioned in A143223. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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..413
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 (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
E. W. Weisstein, Legendre's Conjecture in MathWorld [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 02 2008]
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FORMULA
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A143223(n) < 0
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EXAMPLE
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There are 10 primes between 42 and 2*42, but only 9 primes between 42^2 and 43^2, so 42 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, n]], {n, 0, 500}]; L
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CROSSREFS
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See A000720, A014085, A060715, A143223, A143224, A143225.
Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
Sequence in context: A125009 A008886 A029695 this_sequence A043136 A039313 A043916
Adjacent sequences: A143223 A143224 A143225 this_sequence A143227 A143228 A143229
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KEYWORD
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nonn
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AUTHOR
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Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jul 31 2008
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