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Search: id:A091652
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| A091652 |
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A stable set of primes created by a greedy algorithm. |
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3
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| 3, 7, 13, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 113, 127, 139, 151, 157, 167, 181, 193, 199, 211, 223, 227, 233, 241, 251, 263, 271, 277, 293, 317, 337, 349, 359, 367, 373, 379, 389, 401, 409, 421, 433, 439, 443, 449, 457, 467, 479, 491, 503, 523
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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The Greenfields show that the integers from 1 to 2n can always be paired to form n (not necessarily distinct) primes. A greedy algorithm, starting with 2n, quickly finds the n primes. Interestingly, as n increases, the set of primes produced by this algorithm forms a stable set of prime numbers. Why?
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
L. E. Greenfield and S. J. Greenfield, Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate, J. Integer Sequences, 1998, #98.1.2.
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EXAMPLE
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When the greedy algorithm pairs the numbers 1 to 20, it finds the following 10 primes: 37=20+17, 37=19+18, 31=16+15, 23=14+9, 23=13+10, 23=12+11, 13=8+5, 13=7+6, 7=4+3, and 3=2+1.
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MATHEMATICA
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n=1000; lst=Reverse[Range[2n]]; prms={}; Do[m=lst[[1]]; lst=Delete[lst, 1]; pos=1; While[Not[PrimeQ[m+lst[[pos]]]], pos++ ]; prms=Union[prms, {m+lst[[pos]]}]; lst=Delete[lst, pos], {i, n}]; prms
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CROSSREFS
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Cf. A091653 (complement of these primes), A091654 (frequency of these primes).
Sequence in context: A051336 A002623 A081662 this_sequence A134197 A053001 A053607
Adjacent sequences: A091649 A091650 A091651 this_sequence A091653 A091654 A091655
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KEYWORD
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easy,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jan 26 2004
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