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Old 2021-06-10, 22:32   #1
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×157 Posts
Default 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
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Old 2021-06-14, 01:35   #2
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×157 Posts
Thumbs up small insight

look

That took effort.

Going to go eat now.

Matt
Attached Files
File Type: txt divisor count.txt (956 Bytes, 37 views)
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Old 2021-06-14, 06:39   #3
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×157 Posts
Thumbs up interesting to me

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
Attached Files
File Type: txt more divisor count observation.txt (849 Bytes, 35 views)
File Type: txt Anderson conjecture on divisors.txt (2.2 KB, 38 views)

Last fiddled with by MattcAnderson on 2021-06-14 at 07:45 Reason: another nifty file i typed
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Old 2021-06-14, 16:24   #4
MattcAnderson
 
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"Matthew Anderson"
Dec 2010
Oregon, USA

11101011102 Posts
Smile singly recursive expression b(n) = 2*b(n-1) + 1.

Quote:
Originally Posted by MattcAnderson View Post
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
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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