Paul,
<<What you need is something like
sum(|R(n,n')|; n in [x,y]) i.e. tell us _which |R(n,n')|'s are being
added up. >>
The semi-colon is not a typo? Have only ever encountered colons in formal
expressions.
<<What would make sense is if your t(n,m) was the _set_ of primes less
than or equal to m which are factors of n. You could then reference
|t(n,m)| if you wanted the count of those primes i.e. get your original
t(n,m).>>
Please remember that my *first* objective is to prove propositions that
have
nothing to do with factorisation.
<<With the new t(n,m) you could let R(n,n') = t(n,m) - t(n',m) = the set
of primes in P(m) that are factors of n but not of n' or
R(n,n') = |t(n,m) - t(n',m)| = how _many_ such primes there are or
R(n.n') = |t(n,m)| - |t(n',m)| = how many more (or less) such primes
divide n that divide n'. Which, if any, of those you want I don't know.>>
R(n.n') = |t(n,m)| - |t(n',m)|, as long as it implies that there are
individual members, r, of R(n,n').
<<We shall be constructing a binary r by s matrix M such that
sum(M(i,n); i = 1 .. r) = max(i : M(i,n) =/= 0). [Here, =/= means "not
equal (remember M is binary), M(i,j) is the entry in the i-th row and
j-th column.]>>
Do the r and s here refer to the members r of R(n,n') and s of S(n,n')? If
not, as I strongly suspect, then surely I've got trouble?
<< > Let P(m) be the first m primes.
>
> Let J be a subset of P(m).
>
> Given an interval [x,y] of integers, define N(n) to be the set of
positive
> entries in a column indexed by an integer n in [x,y], N(x,y) to be the
set
> of positive entries in [x,y], and N(x,y,|J|) to be the set of positive
> entries in [x,y] such that the greatest value of |N(n)| in [x,y] is |J|.
Is the matrix still M?>>
Yes.
<< You've got problems : N(n) = {1} or N(n) = {}.>>
The *members* of N(n) are either 1 or 0. My ultimate interest is the
cardinality of N(n).
<<Moreover what if M only has 10 columns and [x,y] = [100,110]?>>
I don't know; how do I get round this? I had learnt previously that one
can
define a matrix's extents and speak quite safely of columns outside of it,
as though it will be protracted ad infinitum.
<<I was guessing you meant x >= 0.>>
Yes.
<<No? If so N(x,y) = y - x + 1 or (in
case x = 0) y-x.>>
But if x=0 and y =10, there are surely 11 columns in the matrix.? I want
zero to be an index value.
<<|N(n)| is either 0 or 1.>>
But I thought I had defined it as the column sum.
<<Suppose M turned out to be
1 0 1 1
0 1 1 0
0 0 1 1
1 1 1 0
What would you want N(1), N(2), N(3) and N(4) to be?>>
{1,1}, {1,1},{1,1,1,1},{1,1} respectively. Cardinalities 2, 2, 4, 2
respectively.
With thanks.
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