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Search: id:A023193
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| A023193 |
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Conjecturally, a(n) is the largest number of primes that occurs on an infinite number of intervals of n consecutive integers. |
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5
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| 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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According to the link at www.opertech.com, a(3159) >= 447 > 446 = pi(3159). The First Hardy-Littlewood conjecture (k-tuples conjecture) then implies that, for an infinitude of n, the interval [n+1, n+3159] includes 447 primes. For these n, pi(n+3159) >= pi(n)+447 > pi(n)+446 = pi(n)+pi(3159), contradicting the Second Hardy-Littlewood conjecture that pi(x+y) <= pi(x)+pi(y). - David W. Wilson (davidwwilson(AT)comcast.net), May 23 2005
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LINKS
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Eric Weisstein's World of Mathematics, Prime k-Tuples Conjecture.
Author?, Title?
T. Forbes, Prime k-tuplets.
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FORMULA
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Conjecturally, a(n) = lim sup pi(x+n)-pi(x), where pi = A000720. This would follow from the k-tuple conjecture. - David W. Wilson (davidwwilson(AT)comcast.net), May 23 2005
a(n) = minimum m such that A008407(m) >= n. [From Max Alekseyev (maxale(AT)gmail.com), Nov 03 2008]
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CROSSREFS
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Cf. A008407 (minimal difference of first and last prime in a prime k-tuplet), A047947 (Schinzel's rhobar), A066081 (weaker binary conjectures).
Least inverse is A020497.
Sequence in context: A024542 A098424 A098428 this_sequence A096605 A109497 A156078
Adjacent sequences: A023190 A023191 A023192 this_sequence A023194 A023195 A023196
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson (davidwwilson(AT)comcast.net)
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