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A120447 a(n) is the multiplier of the primorial which furnishes the "one-line proof" for primality of prime(n) associated with A079900(n); i.e., k*(p+k) = primorial(q)*a(n), where k = A079900(n), p = prime(n) and q = sqrt(p). 0
3, 4, 3, 4, 2, 5, 3, 7, 4, 1, 6, 4, 6, 3, 5, 2, 8, 3, 1, 17, 16, 2, 3, 8, 4, 2, 1, 18, 13, 4, 29, 10, 2, 1, 15, 1, 6, 3, 46, 12, 33, 1, 4, 19, 52, 2, 12, 53, 31, 2, 12, 1, 14, 23, 3, 13, 15, 5, 2, 24, 19, 9, 15, 67, 46, 122, 10, 62, 43, 1, 18, 3, 5, 257, 130, 297, 1, 577, 402, 218, 173, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A one-line proof looks like this: 149 = 5*5*11 - 2*3*3*7. 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. Multiply the left and right products together, then divide out each Q once, the sequence gives the remaining product.

The generating program listed for A079900 tries every k from 1 up until a solution is found. A study of this sequence was used to design a generating program that skips k values where i*Q# < k*(p+k) << (i+1)*Q#.

LINKS

Table of n, a(n) for n=1..82.

FORMULA

a(n) = A079900(n)*(prime(n) + A079900(n))/primorial(sqrt(prime(n))).

EXAMPLE

a(28) = 18 because prime(28) = 107, A079900(28) = 28 and 28*(107 + 28) = 3780 = (2*3*5*7)*18.

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*(p+k)/prod]]]]; a/@Prime/@Range[70]

CROSSREFS

Cf. A079900.

Sequence in context: A090673 A270827 A293072 * A083021 A102745 A108026

Adjacent sequences:  A120444 A120445 A120446 * A120448 A120449 A120450

KEYWORD

nonn

AUTHOR

Bruce A. Leenstra (leenstrb(AT)comcast.net), Aug 24 2006

STATUS

approved

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Last modified October 23 16:20 EDT 2021. Contains 348215 sequences. (Running on oeis4.)