# Duplicate hunting, pt. 10

Martin Fuller martin_n_fuller at btinternet.com
Sun May 6 12:54:26 CEST 2007

```--- Andrew Plewe <aplewe at sbcglobal.net> wrote:
> Possible duplicates:
> A051538 and A119635

They are the same sequence with different offsets:
A051538(n) = A119635(n-1)

Proof:
k, k+1, 2k+1 are co-prime so their lcm is the same as their product.
=> A051538(n) = lcm{k, k+1, 2k+1 | k=1..n}/6

{k, k+1, 2k+1 | k=1..n} = {1..2n+2 excluding even numbers >n+1}
Adding the higher even numbers to the set doubles the LCM.
=> lcm{k, k+1, 2k+1 | k=1..n}/6 = lcm{1..2n+2}/12 = A119635(n-1)

Andrew, Seqfans:

Thanks for finding all these duplicates.  Let me respond:

1. If the A-number of a duplicate is (roughly) >A110000 then
I recycle the A-number when I do the merge.   Otherwise
I merge them and leave the later one marked as "dead".

2. The duplicates involving the cyclotomic polynomials
will be harder to deal with.  But as these sequences
are of little interest I am going to postpone dealing
with them.  I did deal with one pair, two sequences
which differed after a couple of hundred terms, by
giving more terms so that the place where
they first differed was  visible.  That was just one pair
out of a large number.  The others will have to wait.
[One possibiilty is that Simon Plouffe, who first

3.  I am slowly working my way through Andrew's lists
of duplicates (in a fairly random order).
But I currently also have about 350
other comments waiting to be processed, so this will
take some time.

Neil

```