sequences from inequalities

Rainer Rosenthal r.rosenthal at
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
   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:

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

Rainer Rosenthal
r.rosenthal at

More information about the SeqFan mailing list