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MattcAnderson 2021-06-10 22:32

prime divisors
 
Hi again all,

Some of us are familiar with proper divisors.

For example, the proper divisors of 9 are 1 and 3.

Also, the proper divisors of 35 are 5 and 7.

~

Made a Maple procedure

called ProperDivisors(b).

Has count function

Pretty easy to understand

For example -

CPD(6) = 6

That is count proper divisors is 1 and two and three is 6.

Another example

CPD(19) = 1

See my 'blog o ria'

My (big) question is,

Suppose you have a general positive integer

in factored form

call it d.

So d = p1^e1 * p2^e2 * ...

what is its count of proper divisors?

Is there a Maple function?

what is CPD(d) ?

Let me know.

Regards,

Matt

MattcAnderson 2021-06-14 01:35

small insight
 
1 Attachment(s)
look

That took effort.

Going to go eat now.

Matt

MattcAnderson 2021-06-14 06:39

interesting to me
 
2 Attachment(s)
new observation about divisors and positive integers (whole numbers)

an curve fit with recursion namely

b(0)=2 for squares
or
b(0) = 3 for cubes

then

b(n) = 2*b(n-1) + 1.

This data table

b Divisors(b) relevant expression

0 3 3^3
1 7 3^3*5
2 15 3^3*5*7
3 31 3^3*5*7*11


For example Divisors(3) could have relevant expression 7^3*23*29*17.
We see that there is a prime squared followed by three distinct primes.
Then Divisor(3) is 2*15 + 1 which is 31.

Similarly, Divisors(2) could have relevant expression 17^3*3*5
and still Divisors(2) is still 15.

So, in some sense, the primes are interchangable under this 'Divisors count' function.

See you later,
Matt

MattcAnderson 2021-06-14 16:24

singly recursive expression b(n) = 2*b(n-1) + 1.
 
1 Attachment(s)
[QUOTE=MattcAnderson;580922]new observation about divisors and positive integers (whole numbers)

an curve fit with recursion namely

b(0)=2 for squares
or
b(0) = 3 for cubes

then

b(n) = 2*b(n-1) + 1.

This data table

b Divisors(b) relevant expression

0 3 3^3
1 7 3^3*5
2 15 3^3*5*7
3 31 3^3*5*7*11


For example Divisors(3) could have relevant expression 7^3*23*29*17.
We see that there is a prime squared followed by three distinct primes.
Then Divisor(3) is 2*15 + 1 which is 31.

Similarly, Divisors(2) could have relevant expression 17^3*3*5
and still Divisors(2) is still 15.

So, in some sense, the primes are interchangable under this 'Divisors count' function.

See you later,
Matt[/QUOTE]

Today is a new day. I woke up, made my wife's cup, packed her lunch bag, and she is out the door.

Now I do a little Maple Code. I use notepad for the data tables and the insights.

see attached.


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