A079900 - OEIS

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, 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

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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

Table of n, a(n) for n=1..65.

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

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