A002445 question

Dean Hickerson dean at math.ucdavis.edu
Fri Oct 19 07:52:52 CEST 2007

David Wilson wrote:

> What values can occur in A002445?
> Could we write a sorted version of A002445?
> It looks like it might start
> 1,6,30,42,66,138,282,330,354,498,510,642,690,798,870,1002,...
> How could we be sure we had them all?

 From the OEIS entry:

    %N A002445 Denominators of Bernoulli numbers B_2n.
    %C A002445 From the Von Staudt-Clausen theorem, denominator(B_2n) =
               product of primes p such that (p-1)|2n.

Except for a(0)=1, all terms are divisible by 6 and are squarefree.  To
test such a number k to see if it's in the sequence, let 2n be the least
common multiple of all p-1 for which p is a prime divisor of k.  Now list
the primes p such that p-1 divides 2n.  If all of them are divisors of k,
then k is in the sequence; otherwise it's not.

For example, consider  k = 78 = 2 * 3 * 13.  The LCM of 2-1, 3-1, and 13-1
is 12, so 2n=12.  The primes p such that p-1 divides 12 are 2, 3, 5, 7, and
13.  Since 5 and 7 aren't divisors of 78, 78 is not in the sequence.

Dean Hickerson
dean at math.ucdavis.edu

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