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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[70]
CROSSREFS
Sequence in context: A003417 A358549 A158986 * A188317 A117354 A140324
KEYWORD
nonn
AUTHOR
Don Reble, Feb 20 2003
EXTENSIONS
Edited by Dean Hickerson, Feb 24 2003
STATUS
approved

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Last modified March 3 04:18 EST 2024. Contains 370499 sequences. (Running on oeis4.)