# sequences from inequalities

Rainer Rosenthal r.rosenthal at web.de
Sat Sep 4 00:45:08 CEST 2004

```> Me:  let me remind everyone that the correct thing
> for seqfans to do here is to convert these inequalites
> into sequences, and then send them in to the OEIS if
> they are not there!

Aye, Aye Sir!

"The product of consecutive numbers is never a square?" is a
thread in sci.math these days. I tried to reformulate this
in a way, which would produce a sequence:

For integer k define
SquareMod(k) = k - q
for the largest square
q less or equal to k.

Example: SquareMod(33) = 33-25 = 8. (See (*) below.)

Let P be a product of n consecutive integers. If P is not
a square we get SquareMod(P) > 0. I puzzled a little to
get a sequence definition from this.

1) The easiest way was to consider P = n! (factorial).
So  I was lead to a(n) = SquareMod(n!).
Computing the first terms revealed it as
http://www.research.att.com/projects/OEIS?Anum=A066857
0 1 2 8 20 44 140 320 476 3584 12311 4604 74879.
^^^^^

2) A more advanced and sort of thrilling way is to consider
*all* products P > 0 of n consecutive numbers, defining

a(n) = minimum of all SquareMod(P), where
P > 0 is product of n consecutive integers

It *seems* as if it starts as follows:
0 1 2 8 20 44 140 320 476 3584 4604 4604 74879
^^^^

Most of the the entries are just a(n) = A066857(n).
But there is an exception: a(10) = a(11) because 12! is
closer to the next square below than 11!

(*) I detected my SquareMod(n) as "square excess" in the OEIS:
http://www.research.att.com/projects/OEIS?Anum=A053186

Considering 2) I am completely uncertain about the values
given. And there may be many more exceptions. Who knows?

And now the SeqFan question: how to deal with such an unreliable
sequence?

Rainer Rosenthal
r.rosenthal at web.de

```