[seqfan] Re: %C A170826

[Maximilian Hasler]

On Mon, Dec 28, 2009 at 9:56 AM, zak seidov <zakseidov at yahoo.com> wrote:
> A170826 1,2,3,8,5,36,7,64,81,100,11,144,13,196,225,256,17,324,
> %N A170826 a(n) = GCD(n^2, n!).
>
> %C A170826 If n is prime then a(n)=n else if n<>4 a(n)=n^2(due to The Fundamental Theorem of Arithmetic).

You don't need the fundamental thm here, just by definition,
n is not prime <=> there are a,b < n such that n=ab.
If n is not the square of a prime, then we can assume a<b,
so both these a,b occur "explicitely" in n!, as does n itself.

If n is the square of a prime > 2 then there exists another multiple
of that prime (e.g. 2p) less than n, which occurs explicitely in n!

So for all composite n  except for n=2x2=4,  n! is always a multiple of n^2.

Maximilian