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Search: id:A161622
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| A161622 |
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Denominators of the ratios (in lowest terms) of numbers of primes in one square interval to that of the interval and its successor. The numerators are derived from sequence A014085. The expression is: R(n)=(PrimePi[(n+1)^2] - PrimePi[n^2])/(PrimePi[(n+2)^2] - PrimePi[n^2]); The first few ratios are: 1/2, 2/5, 3/5, 1/3, 4/7,... |
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+0
5
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| 2, 2, 5, 5, 3, 7, 7, 7, 8, 9, 9, 2, 9, 5, 13, 12, 11, 2, 13, 2, 2, 13, 5, 17, 15, 15, 17, 17, 2, 9, 19, 19, 19, 19, 19, 2, 7, 23, 23, 23, 20, 7, 23, 24, 23, 23, 28, 5, 21, 26, 31, 7, 25, 24, 23, 29, 30, 29, 2, 29, 30, 32, 29, 15, 31, 2, 32, 30, 34, 12, 2, 32, 2, 35, 20, 18, 16, 41, 36, 33
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Conjecture: Lim R(n) as n->oo = 1/2. See also more extensive comment entered with sequence of numerators. This conjecture implies Legendre's conjecture.
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EXAMPLE
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R(3) = (PrimePi(4^2)-PrimePi(3^2)) / (PrimePi(5^2)-PrimePi(3^2)) = (PrimePi(16)-PrimePi(9)) / (PrimePi(25)-PrimePi(9)) = (6-4)/(9-4) = 2/5. Hence a(3) = 5. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
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PROGRAM
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(MAGMA) [ Denominator((#PrimesUpTo((n+1)^2) - a) / (#PrimesUpTo((n+2)^2) - a)) where a is #PrimesUpTo(n^2): n in [1..80] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
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CROSSREFS
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A014085
Cf. A161621 (numerators). [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009]
Sequence in context: A054079 A005177 A045537 this_sequence A116559 A008280 A063960
Adjacent sequences: A161619 A161620 A161621 this_sequence A161623 A161624 A161625
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KEYWORD
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nonn
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AUTHOR
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Daniel Tisdale (daniel6874(AT)gmail.com), Jun 14 2009
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EXTENSIONS
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a(1) inserted and extended beyond a(11) by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 15 2009
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