Volume 9, Issue 4
|
Articles
Tricks of the Trade
In and Out
Trott's Corner
New Products
New Publications
Calendar
News Bulletins
New Resources
Classifieds
Download This Issue
Editorial Policy
Staff and Contributors
Submissions
Subscriptions
Advertising
Back Issues
Contact Information
   |
k-ary Divisors
Q: The 
-ary divisors of 
, written 
, are defined at mathworld.wolfram.com/k-aryDivisor.html. How can I implement 
?
A: A divisor of a number 
is a number 
which divides 
(written 
), also called a factor. The divisors 
of 
, that is 
, are computed using Divisors. Here are the divisors of 12.
It is straightforward to implement the notation 
.
Clearly 3 divides 12.
A 1-ary or unitary divisor (mathworld.wolfram.com/UnitaryDivisor.html) 
of 
satisfies 
, that is 
is relatively prime to 
or 
. divisors[1] [n] computes, and caches, the list of unitary divisors of 
, 
.
Here are the unitary divisors of 12.
Define the notation 
.
In this notation, a divisor 
of 
is written 
.
Here are the unitary divisors for the first few integers (Sloane's A077610).
For a prime power 
, the unitary divisors are 1 and 
[1].
2 is a divisor but not a unitary divisor of 12.
is called a 
-ary divisor of 
, written 
, if the greatest common 
-ary divisor of 
and 
is 1. Recursive implementation, using caching, is straightforward.
Here are the 2-ary or biunitary divisors for the first few integers.
For a prime power 
, the biunitary divisors are 
except for 
when 
is even [1].
Here are the 
-ary divisors of 
for 
. One observes that a fixed point is attained.
A divisor 
of 
is called infinitary, 
, if it is a product of divisors of the form 
, where 
is a prime power dividing 
and 
is the binary representation of 
(see Sloane's A037445). The infinitary divisors are then those factors 
that have zeros in the binary representation of all 
where 
itself does. This can be directly coded using BitOr.
is an infinitary divisor of 
(with 
) if 
[1].
Here are the infinitary divisors for the first few integers.
Taking the logarithm base 
of the infinitary divisors of 
gives the 
values for which 
are infinitary divisors of 
.
Here is a plot of the infinitary divisors of 
, computed with 
but valid for arbitrary 
, for 
. Showing the last (decimal) digit of 
enables the actual divisors to be read off easily [1].
The appearance of the Sierpinski sieve fractal mathworld.wolfram.com/SierpinskiSieve.html is interesting.
|
About Mathematica | Download Mathematica Player
© 2005 Wolfram Media, Inc. All rights reserved.