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 A079900 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). 1
 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; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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 Donald S. McDonald and David W. Wilson. REFERENCES R. K. Guy, Lacampagne and J. Selfridge, Primes at a glance, Math Comput 48(1987) 183-202; Math. Rev. 87m:11008. LINKS EXAMPLE a(6)=2: The 6th prime is 13 and the equation 13 = 3*5 - 2 proves it. MATHEMATICA 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 CROSSREFS Sequence in context: A083531 A003417 A158986 * A188317 A117354 A140324 Adjacent sequences:  A079897 A079898 A079899 * A079901 A079902 A079903 KEYWORD nonn AUTHOR Don Reble, Feb 20 2003 EXTENSIONS Edited by Dean Hickerson, Feb 24 2003 STATUS approved

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Last modified April 14 07:59 EDT 2021. Contains 342946 sequences. (Running on oeis4.)