In article <AkgUj.2943$Lq4.1773@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
[...]
> My ultimate goal is to show how, if such-and-such values (in my
definitions
> given above) are held fixed and such-and-such values change, then
o(x,y,m),
> as defined above, changes in such-and-such a manner.
> At present I don't know whether I am going to have to have a whole lot
of
> other definitions for my array on top of the ones I have laid out above,
> when basically both sets of deinitions are defining the same thing, save
> that in the one case there is the constraint of the t(n,m) being the
number
> of factors of n in J and in the other, there is no such constraint.
[...]
A few general comments.
First, are you familiar with Legendre's identity? Let P be a product of
distinct primes (like your a(m)) and let A be a set of integers ( like
your [x, y] ). Lengendre's identity is a formula (sort of) for the
number of integers in A which share _no_ primes with P. It seems to me
that your o(x,y,m) is exactly y - x + 1 minus that number.
The reason for my M(J,C) instead of your M_1 is that it is frequently
better to start with a general definition and then specialize (kind of
like C+ programming). You con always let J and C be anything you want.
For example you can let P the first m primes in increasing order and C
be [x, y] in increasing order; M(P,C) will then tell you about the
divisibility of elements of [x,y] by one of the first m primes. If you
wanted to change the interval you could just let C' = [x',y'] etc.
Using 0's and 1's has the advantage of letting you use logical
operations; for instance if you OR the columns of M(P,C) the result
will have 1's in exactly the rows corresponding to primes which divide
at least one of the elements of C.
Your paragraph quoted above is a little too vague for me to be of any
help. Are you going to change m's or intervals or both or none of the
above. How many changes at one time are you going to want to make?
--
Paul Sperry
Columbia, SC (USA)
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