

A079900


a(n) = the smallest positive number which furnishes a "oneline proof" for primality of prime(n), the nth 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
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OFFSET

1,6


COMMENTS

A oneline proof looks like this: 101 = 2*3*3*7  5*5. For each prime Q up to the squareroot 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.


REFERENCES

R. K. Guy, Lacampagne and J. Selfridge, Primes at a glance, Math Comput 48(1987) 183202; 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: A083531 A003417 A158986 * A188317 A117354 A140324
Adjacent sequences: A079897 A079898 A079899 * A079901 A079902 A079903


KEYWORD

nonn


AUTHOR

Don Reble (djr(AT)nk.ca), Feb 20 2003


EXTENSIONS

Edited by Dean Hickerson, Feb 24 2003


STATUS

approved



