20210610, 22:32  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
1756_{8} 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 
20210614, 01:35  #2 
"Matthew Anderson"
Dec 2010
Oregon, USA
2×503 Posts 
small insight
look
That took effort. Going to go eat now. Matt 
20210614, 06:39  #3 
"Matthew Anderson"
Dec 2010
Oregon, USA
2·503 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(n1) + 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 Last fiddled with by MattcAnderson on 20210614 at 07:45 Reason: another nifty file i typed 
20210614, 16:24  #4  
"Matthew Anderson"
Dec 2010
Oregon, USA
2×503 Posts 
singly recursive expression b(n) = 2*b(n1) + 1.
Quote:
Now I do a little Maple Code. I use notepad for the data tables and the insights. see attached. 

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