

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.


REFERENCES

R. K. Guy, Lacampagne and J. Selfridge, Primes at a glance, Math Comput 48(1987) 183202; 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



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



