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Search: id:A143224
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| A143224 |
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Numbers n such that (number of primes between n^2 and (n+1)^2) = (number of primes between n and 2n). |
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10
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| 0, 9, 36, 37, 46, 49, 85, 102, 107, 118, 122, 127, 129, 140, 157, 184, 194, 216, 228, 360, 365, 377, 378, 406, 416, 487, 511, 571, 609, 614, 672, 733, 767, 806, 813, 863, 869, 916, 923, 950, 978, 988, 1249, 1279, 1280, 1385, 1427, 1437, 1483, 1539, 1551, 1690
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The sequence gives the zeros in A143223. The number of primes in question is A143225(n).
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.
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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.
S. Ramanujan, Collected Papers of Srinivasa Ramanujan (G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson, eds.), Amer. Math. Soc., Providence, 2000, pp. 208-209. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
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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 (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]
S. Ramanujan, A Proof Of Bertrand's Postulate [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
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FORMULA
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A143223(n) = 0
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EXAMPLE
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There are the same number of primes (namely 3) between 9^2 and 10^2 as between 9 and 2*9, so 9 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, 2000}]; L
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CROSSREFS
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See A000720, A014085, A060715, A143223, A143225, A143226.
Cf. A104272, A143227. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 03 2008]
Sequence in context: A118414 A137628 A020297 this_sequence A068810 A077115 A073946
Adjacent sequences: A143221 A143222 A143223 this_sequence A143225 A143226 A143227
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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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