[seqfan] Re: Close sequences

[Maximilian Hasler]

On Sat, Oct 17, 2009 at 6:30 PM, Christopher Gribble
<chris.eveswell at virgin.net> wrote:
> I am considering submitting the following sequence:
> 0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,

It seems this has already been submitted 3 days ago.

> For which the definition is:
> a(n) = minimum value of j, 1 <= j <= n-1, such that floor(j^2 / n) > 0 for each n.

If I understand correctly the meaning of the "for each n", then it
should simply be omitted.

I don't understand why you say  1<=j<=n-1 , it seems irrelevant except
for n=1 where it produces an empty set of possible j's, which has
+infinity as infimum.
Therefore I also don't understand a(1)=0, it does not verify j >= 1
nor  floor(..)>0.

And a(n)=2 also seems incompatible with 1<=j<=n-1.

I think the condition "j <= n-1" is highly artificial
(your triangle A166373 would look just as nice if it had one more
element in each row)
and this is the only reason for which this sequence is not identical to A003059.

For all other terms (i.e. n>1), your definition is equivalent to
min { j | [j^2/n] > 0 } = min { j | j^2 >= n } = ceil( sqrt( n )) = A003059(n).

Maximilian