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 2021-06-10, 22:32 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 2×3×157 Posts 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
2021-06-14, 01:35   #2
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×157 Posts
small insight

look

That took effort.

Going to go eat now.

Matt
Attached Files
 divisor count.txt (956 Bytes, 37 views)

2021-06-14, 06:39   #3
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

2×3×157 Posts
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
 more divisor count observation.txt (849 Bytes, 35 views) 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

2021-06-14, 16:24   #4
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

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

Quote:
 Originally Posted by MattcAnderson 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.
Attached Files
 proper divisors at one level recurrsin and a data table for Mersenne number.txt (432 Bytes, 32 views)

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