Paul,
>> 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?
>
Yes.
> 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).
>
Indeed on my reader they don't line up properly; but can I take it there
are
two X's at n=4 and n=8? In M_1, there would be one for each, as there is
only one prime factor, 2. So in my definition <<For an integer n let
t(n,m)
be the number of members of P(m) that divide n>>, would you be able to
advise me on phrasing?
> 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.
Yes, that's just it.
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?
It might very well be the case. As I say, I feel my matrix M_1 will
become
redundant. But, Paul, I hope we don't lose sight of what I am really
driving
at, which is first and foremost an issue of notation. I want to be able to
use the concepts -- column sum etc. -- behind all those definitions that I
put forward - t(n,m), o(x,y,m) etc. not only for M_1 but also for the kind
of histogram/array we have discussed. I am very keen to know how to lay
out
the definitions. I'm told I mustn't use the same definitions for two
different arguments, which makes me think the whole thing is going to be
inefficient in terms of use of letters of the alphabet. Perhaps it would
be
best just to define a set, B, of entries, whose members are b, then to use
\sum_b(n) in place of t(n,m), 1/2*sum_b(n) *((sum_b(n))-1) for c(n,m)
etc..
It's just it looks very ***bersome; and it will get considerably worse
when
I start referring to the set T(x,y,m). Maybe, for my matrix as distinct
from
my array/histogram, I simply use the same terms but with a subscript?
I've got a hunch that you will object to my use of 'entries', above, in
relation to a histogram. Let's imagine what I term 'entries' as units in a
histogram bin. One thing that is crucial to my method is a system of
forming
pairs of units, each pair comprising one unique unit from a bin in an
interval [x,y] and one unique unit from a different bin in a different
interval [x',y']. The reason I was reluctant immediately to construct
histograms instead of what I have been referring to as 'arrays' was that I
guessed that amathematician would regard the bin value as just a number:
it's presumably not to be regarded as a set of elements that can be paired
in the manner described. This is something that troubles me, as I don't
know
the relevant conventions. My problem is, as ever, one of expressing my
final
version in a professional manner.
With many thanks once more for your continuing help.
|
