Paul,
It looks like we're making good progress. Comments embedded.
> I've paraphrased your definitions below. I saw nothing wrong with them
> the way they were; my changes just make typing them easier and give you
> a little more flexibility if you ever want it.
>
> <<Let P(m) be the set of the first m prime numbers.
> Let a(m) be their product.
>
> My changes start here.
>
> Let P be a set of primes and let I be a set of positive integers.
> [Anytime you want to you can insert the statement "Let P = P(m) and
> I = [x,y]" or words to that effect.]
>
> For an integer n let t(n,P) be the number of members of P that divide n.
But, im****tantly, we are not just taking an arbitrary set of primes. In
P(m)
we are taking those primes that divide multiples, of value less than the
square of (y-x+1). Surely it's easier just to begin with a set P(m) and
define a subset, J?
>
> Define c(I,P) to be the sum of (1/2)t(n,P)(t(n,P)-1) for n in I.
> [Or, sum((1/2)t(n,P)(t(n,P)-1); n in I)]
>
> Define T(I,P) to be the set n in I for which t(n,P) > 1.
>
> Define o(I,P) to be the number of n in I for which t(n,P) > 0.
>
> Define N(I,P) to be the sum of t(n,P) for n in I.
> [Or, sum(t(n,P); n in I)] >>
>
> As I said, once you say P = P(m) and I = [x,y] these are exactly what
> you had - just easier to type.
If we go back to the problem that I'm really keen to resolve, which is how
I
use both an array and the matrix M_1 in the way I have already discussed,
then what do you think I should do? Maybe for the array, use a subscript
eg.
T_{\alpha}(I) and for the matrix, use your T(I,P)? Note that I omit the
reference to P in the array, because the entries in the array are not
defined by divisibility; but there will come a time when I want the
maximum
possible column sum in the array to be #P. I suppose in that case I simply
write T_{\alpha}(I,#P)?
There is then the other issue that you object to my use of the term array;
so perhaps you are advising me to construct a matrix M_[x] instead, in
which
M_[x] has all the characteristics of what I am calling an array?
>
> There is something called the Cartesian (or "cross") product of sets.
> (Forgive me if you already know all about this.)
I have heard mere mention of something along these lines.
If A and B are sets
> then A x B is the set of all ordered pairs (a,b) where a is in A and b
> is in B. Read A x B as "A cross B". "Ordered" means, for example that
> (2,3) is different from (3,2).
So I can say [x,y]*[x',y']? I wonder if you could fill me in with a
precise
wording, given how I have described by objectives - that would be much
appreciated.
In making my pairs (n,n'), I am trying to construct a set of sets R(n,n')
such that R(n,n') = t(n,m)-t(n',m), and, for all R(n,n'),
t(n,m)-t(n',m)>0,
and sum_R(n,n') is as low a value as it possibly can be given [x,y] and
[x',y'].
>
> So, if you wanted to pair up elements of I = [x,y] with elements of
> I' = [x',y'], you would just write I x I'. [For example,
> sum(t(i,P)*t(j,P); (i,j) in I x I') would sum over all possible
> products with i in [x,y] and j in [x',y']. I don't know why you would
> want to do that but you _could_.] If you only want some of the pairs
> there is nothing wrong with using a subset of I x I'. For example you
> could let B be the set of all elements (a,b) of I x I' such that a < b.
>
> I didn't understand your use of "bin" so I just ignored it :-)
I had understood a bin to be a column in a histogram.
With many thanks.
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