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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: A024680 A083531 A003417 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(AT)math.ucdavis.edu), Feb 24 2003
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