Paul,
Thanks once more. Comments embedded.
..
>
> Well, there's the rub: what _is_ the issue you have? In particular, it
> is im****tant (for me) to know if you are intending the "matrix" as
> merely a data structure rather than a mathematical object; i.e. are you
> planning on doing any matrix algebra?
At present I don't think I necessarily need it in my final paper. But I
was
discussing it here because I am entertaining the idea of merging two
constructs - an array and a matrix - into one. So I am just using it to
explain my issue of notation.
>
>> Currently, I have got the following:
>>
>> <<Let P(m) be the set of the first m prime numbers.
>> Let a(m) be their product. For an integer n let t(n,m)
>> be the number of members of P(m) that divide n (no
>> multiplicity counted, so t(9,2) = 1, for example).
>>
>> Given an interval [x,y] of integers,
>> define c(x,y,m) to be the sum of
>> (1/2)t(n,m)(t(n,m)-1) for n in [x,y].
>>
>> Given an interval [x,y] of integers, define
>> T(x,y,m) to be the set n in [x,y] for which
>> t(n,m) > 1.
>>
>> Given an interval [x,y] of integers, define
>> o(x,y,m) to be the number of n in [x,y] for
>> which t(n,m) > 0.
>>
>> Given an interval [x,y] of integers, define
>> N(x,y,m) to be the sum of t(n,m) for n in [x,y].>>
>
>
> Is what you are after the number of distinct prime divisors of an
> integer n?
Not necessarily. You see, I have two distinct objectives. The first is to
show what happens when certain values are held fixed, and some other set
of
values change. For this I am constructing an array. In establi****ng those
principles, I do not want the complication of there being any prescribed
determinant of the number of black cells in a column.
The second objective is to apply the findings of the above to sieves as
defined in my matrix M_1, in particular with regard to such values as
t(n,m). The value of t(n,m) is not necessarily the number of distinct
factors of an integer; it's the number of its distinct factors that are in
J.
>
> The phrase "the number of black cells is not necessarily determined by
> factorisation" is bothersome. What _does_ determine the number of black
> cells?
As I say above, the whole point is that there is no prescribed determining
criterion. It's entirely arbitrary.
For that matter, why the very unmathematical "black" and
> "white"?
>
If you have any suggestions... maybe 'occupied/unoccupied cells'?
>> I was thinking that maybe I could make my array one in which the
numbers
>> of
>> black cells in a column over the first (a(m)+y-x ) columns are
precisely
>> equal to the values found for the matrix we are defining. That way I
>> could
>> make my reference something like t(n,m,[1,a(m)+y-x]) to indicate that n
>> in
>> [x,y] is a column in that interval,
> I can't comment on this since I don't know what the objective is.
My question is ultimately one of notation. I have the concepts in my head,
I
just don't know which ones it is acceptable to mix together in my
definitions.
> Observe however that the sequences J and C can be anything you want
> them to be within the given constraints - J could be all primes less
> than sqrt(n) and C could be the single integer n, for example. A fairly
> common usage is something like "Let M(J,C) = (m(i,j))" and then use
> m(i,j) to refer to the (i,j) entry of M(J,C).
>
> What is your ultimate goal? Just an outline or abstract would help a
> lot. It is looking more and more like it is some sort of integer
> factoring algorithm perhaps using some sort of sieve.
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.
With thanks.
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