Paul,
>>> Which you didn't object to.
>
I am not always sure I understand - and often I flatly don't understand
the
expressions you have put forward, and on the whole I have mentioned where
I
don't understand.
> _Now_ you say
>>>No, I meant *multiples* of a(m) [the product of the first m primes].
That's consistent with my original objectives. I am only using the first m
primes to index rows; so I am building the sieve of Eratosthenes and
extending it without bringing in the new rows that would appear in the
sieve
of Eratosthenes as n increases to being equal to or greater than the
square
of the (m+1)-th prime. Every multiple of a(J) has #J entries and every
multiple of a(m) has #P(m) entries. I also want #J entries at the column
indexed by zero. I am not going to change this method of construction of
M_1
and I never have changed it. That's OK by you isn't it? (The sole
complication is, I have been entertaining the idea of making a hybrid of
M_1
and the array comprising f columns.)
>>>Or a(J) when I am using the subset of [P(m)], J.
>>>The product of all primes in J is a column in which there are
>>> entries.
>>>Same with the column indexed by twice the product, etc. The fact
>>>that these
>>>columns are found exclusively at intervals of [a(m)] is the
>>>reason I want
>>>one such at the column indexed by zero.
>
> Well, I'm lost. How does this conform to your previous definition of
> M_1? For that matter what do you mean by "The product of all primes in
> J is a column in which there are J entries"?
Sorry, I meant #J entries.
>
> In an effort to clear things up I asked for an example. I _still_ would
> like to see an example with actual numbers and matrices/arrays.
For my matrix, the integer on the left indexes the n-th column and the
integer on the right isthe number of entries in that column:
0 #J
1 0
2 1
3 1
4 1
5 1
6 2
7 1
8 1
9 1
10 2
11 1
12 2
>
> You still haven't said what you mean by "array" - it is not a commonly
> used mathematical term.
>
I got it from a retired academic who has lectured full-time in
mathematics;
but I'll take your word for it. I think what he meant was a matrix in
which
the values on the vertical axis did not index values to which there was
associated any specific criterion determining the address of an entry. But
my purposes require something even looser; as I said it might just as well
be a histogram. Take the following; the column numbers, which you could
call
values of n, are given as the bottom row and arbitrary column sums are the
upper row:
4 7 2 0 1 0 5 8 11 1
1 2 3 4 5 6 7 8 9 10
Very simple. As I say, I am not imposing any criterion determining column
sum.
With thanks.
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