I believe that I committed a mistake on having

compiled (I use MATHEMATICA) the functions of the

second identity.

I am afraid that this second identity is not true.

I wait for your commentaries on the first and third

identity.

Sincerely

Sebastián Martin Ruiz

--- David Cleaver <

wraithx@...> escribió:

>

>

> Note:

> LHS stands for Left Hand Side

> RHS stands for Right Hand Side

>

> Sebastian Martin wrote:

> >

> >

> > Dear colleagues:

> >

> > I have found some identities:

> >

> > 1) Product[k^Floor[n/k],{k,1,n}]=

> > Product[Floor[n/k]!,{k,1,n}]

>

> I was able to confirm this for n=5 and n=6.

>

> >

> > 2) Product[Floor[n/k]^k,{k,1,n}]=

> > Product[k!,{k,1,n}]

>

> I was unable to confirm this, ie:

>

> For n = 5

> LHS = 20

> RHS = 34560

>

> For n = 6

> LHS = 432

> RHS = 24883200

>

> Did you perhaps type the wrong formula in?

>

> >

> >

>

3)(Product[(i^j)*(j^i),{i,1,n},{j,1,n}]/Product[i!*j!,{i,1,n},{j,1,n}])^(1/n)=

> > Product[C(n,i),{i,0,n}]

>

> I confirmed this for n = 5.

> (The products in the LHS grow very rapidly!)

> (For n=5 the Numerator was 2.37*10^65 and the

> Denominator was 2.43*10^45)

>

> >

> > OEIS A001142

> >

> > This identities are known?

> >

> > Sincerely

> >

> > Sebastian Martin Ruiz

>

> Good job on finding those identities, I personally

> do not know if they

> have been found before or not, someone else will

> have to speak to that.

> I just wanted to let you know about the problem

> with the 2nd Identity

> that you found.

>

> -David C.

>

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