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Search: id:A079900
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| A079900 |
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a(n) = the smallest positive number which furnishes a "one-line proof" for primality of prime(n), the n-th prime; i.e. the smallest k which is relatively prime to p such that k*(p+k) is divisible by every prime less than sqrt(p), where p=prime(n). |
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+0
2
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| 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 5, 3, 4, 2, 3, 7, 21, 9, 3, 34, 32, 5, 7, 16, 8, 4, 2, 28, 21, 7, 203, 100, 28, 15, 126, 14, 63, 35, 253, 520, 910, 105, 264, 665, 1155, 165, 504, 1155, 858, 156, 495, 91, 539, 715, 198, 507, 550, 275, 143, 720, 627, 2002, 2618, 5695, 4692
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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A one-line proof looks like this: 101 = 2*3*3*7 - 5*5. For each prime Q up to the square-root of p(n), either the left product or the right product is divisible by Q, but not both. It follows that the difference is not divisible by any such Q and so is prime. The sequence gives the right (smaller) number.
The idea comes from seqfan postings by Don McDonald and David W. Wilson.
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REFERENCES
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R. K. Guy, Lacampagne and J. Selfridge, Primes at a glance, Math Comput 48(1987) 183-202; Math. Rev. 87m:11008.
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EXAMPLE
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a(6)=2: The 6-th prime is 13 and the equation 13 = 3*5 - 2 proves it.
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MATHEMATICA
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a[p_] := Module[{prod, k}, prod=Times@@Prime/@Range[PrimePi[Sqrt[p]]]; For[k=1, True, k++, If[GCD[p, k]==1&&Mod[k*(p+k), prod]==0, Return[a[p]=k]]]]; a/@Prime/@Range[70]
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CROSSREFS
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Sequence in context: A083531 A003417 A158986 this_sequence A117354 A140324 A010250
Adjacent sequences: A079897 A079898 A079899 this_sequence A079901 A079902 A079903
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KEYWORD
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nonn
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AUTHOR
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Don Reble (djr(AT)nk.ca), Feb 20 2003
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EXTENSIONS
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Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Feb 24 2003
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