In article <nBzWj.7602$_K5.5388@[EMAIL PROTECTED]
>, Jack <jj@[EMAIL PROTECTED]
>
wrote:
[...]
> > 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;
Nevertheless, if I don't know what it means (and from the next
sentence, neither do you) it is a safe bet that your intended readers
won't know either.
> 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.
Not so simple. Do you mean that in the upper row the numbers were
selected more or less at random?
Maybe I've got it. Maybe you really _do_ want a histogram of sorts.
Depending on your news reader, things may not line up properly - it
looks good in 12pt Monaco.]
|
|
| X
| X X X
| X X X X X X X X X X
__________________________________ ...... ____
1 2 3 4 5 6 7 8 9 10 30
Where the X's indicate the number of primes in the set of the first 4
primes that are factors of the number on the horizontal axis. Is this
"array" of X's by any chance what you had in mind for M_1?
You previously defined t(n,m) to be the count of the number of primes
in the first m primes that are factors of n. The above histogram is
basically a graph of that function t(n,4).
You also had o(x,y,m) which was the count of the number of integers n,
x <= n <= y which have at least one of the first m primes as a factor;
that just gives you a single number. There is, however, nothing to
prevent you from having a sequence of intervals and plotting the values
of o(_,_,m) against those intervals.
If I've finally managed to guess correctly, what you are really
interested in is t(n,m) and o(x,y,m) and the business with the
"matrices" and "arrays" is only a way to visualize the values of those
functions. Also, it occurs to me that you are not using the words
"matrix" and "array" in a precise mathematical sense but rather a loose
English language sense.
Am I getting close?
--
Paul Sperry
Columbia, SC (USA)
|
