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A014085
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Number of primes between n^2 and (n+1)^2.
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70
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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. [From Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 30 2008 [Comment corrected by Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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. [From T. D. Noe (noe(AT)sspectra.com), Sep 05 2008]
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REFERENCES
| J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
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LINKS
| 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
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FORMULA
| a(n) is the number of occurrences of n in A000006 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 17 2003
Pi((n+1)^2)-Pi(n^2) = A000720((n+1)^2)-A000720(n^2). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 30 2008
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EXAMPLE
| 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
| Table[ct = PrimePi[(k + 1)^2] - PrimePi[k^2], {k, 0, 80}]. - Lei Zhou (lzhou5(AT)emory.edu), Dec 01 2005
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PROG
| (PARI) a(n)=primepi((n+1)^2)-primepi(n^2) \\ Charles R Greathouse IV, Jun 15 2011
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CROSSREFS
| Cf. A000006, A053000, A053001, A007491, A077766, A077767, A108954, A000720, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Sequence in context: A126336 A134446 A125749 * A171239 A029210 A035433
Adjacent sequences: A014082 A014083 A014084 * A014086 A014087 A014088
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KEYWORD
| nonn,nice
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AUTHOR
| Jonathan Wild (jon(AT)sound.music.mcgill.ca)
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