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A014085
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Number of primes between n^2 and (n+1)^2.
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89
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0, 2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, 8, 7, 8, 9, 8, 8, 10, 9, 10, 9, 10, 9, 9, 12, 11, 12, 11, 9, 12, 11, 13, 10, 13, 15, 10, 11, 15, 16, 12, 13, 11, 12, 17, 13, 16, 16, 13, 17, 15, 14, 16, 15, 15, 17, 13, 21, 15, 15, 17, 17, 18, 22, 14, 18, 23, 13
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OFFSET
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0,2
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COMMENTS
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Suggested by Legendre's conjecture (still open) that for n > 0 there is always a prime between n^2 and (n+1)^2.
See the additional references and links mentioned in A143227. [Jonathan Sondow, Aug 03 2008]
Legendre's conjecture may be written pi((n+1)^2) - pi(n^2) > 0 for all positive n, where pi(n) = A000720(n), [the prime counting function]. - Jonathan Vos Post, Jul 30 2008 [Comment corrected by Jonathan Sondow, Aug 15 2008]
Legendre's conjecture can be generalized as follows: for all integers n > 0 and all real numbers k > K, there is a prime in the range n^k to (n+1)^k. The constant K is conjectured to be log(127)/log(16). See A143935. [T. D. Noe, Sep 05 2008]
For n > 0: number of occurrences of n^2 in A145445. - Reinhard Zumkeller, Jul 25 2014
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REFERENCES
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Octavian Cira, Smarandache's Conjecture on Consecutive Primes, International J.Math. Combin. Vol. 4 (2014), 69-91; http://mathcombin.com/upload/file/20150127/1422320940239094100.pdf#page=74
J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
Tsutomu 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)
Eric Weisstein's World of Mathematics, Legendre's Conjecture
Wikipedia, Legendre's conjecture
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FORMULA
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a(n) is the number of occurrences of n in A000006 . - Philippe Deléham, Dec 17 2003
pi((n+1)^2) - pi(n^2) = A000720((n+1)^2) - A000720(n^2). - Jonathan Vos Post, Jul 30 2008
a(n) = sum (A010051(k) : k = n^2 .. (n+1)^2). [Reinhard Zumkeller, Mar 18 2012]
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EXAMPLE
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a(17) = 5 because between 17^2 and 18^2, i.e. 289 and 324, there are 5 primes (which are 293, 307, 311, 313, 317).
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MATHEMATICA
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Table[PrimePi[(n + 1)^2] - PrimePi[n^2], {n, 0, 80}]. (* Lei Zhou, Dec 01 2005*)
Differences[PrimePi[Range[0, 90]^2]] (* Harvey P. Dale, Nov 25 2015 *)
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PROG
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(PARI) a(n)=primepi((n+1)^2)-primepi(n^2) \\ Charles R Greathouse IV, Jun 15 2011
(Haskell)
a014085 n = sum $ map a010051 [n^2..(n+1)^2]
-- Reinhard Zumkeller, Mar 18 2012
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CROSSREFS
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Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954, A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A010051.
Cf. A061265, A038107, A221056.
Cf. A000290, A145445.
Counts of primes between consecutive higher powers: A060199, A061235, A062517.
Sequence in context: A126336 A134446 A125749 * A248891 A171239 A029210
Adjacent sequences: A014082 A014083 A014084 * A014086 A014087 A014088
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KEYWORD
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nonn,nice
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AUTHOR
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Jon Wild
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STATUS
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approved
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