terminology for square roots

[franktaw at netscape.net]

Let me expand on the inner and outer square root idea.  If n = Product 
p_i^e_i, then the true square root is Product p_i^(e_i/2).  The inner 
square root is Product p_i^floor(e_i/2), while the outer square root is 
Product p_i^ceiling(e_i/2).  And the inner square root times the outer 
square root is the original number.

Similarly, we can define 3 integer cube roots: the inner cube root 
(A053150) is Product p_i^floor(e_i/3), the central cube root (not in 
OEIS) is Product p_i^round(e_i/3), and the outer cube root (A019555) is 
Product p_i^ceiling(e_i/3).  The product of all 3 cube roots is the 
original number.

In general, there are m integer mth roots of n, indexed by k with 0 <= 
k < m, as Product p_i^floor((e_i+k)/m); and the product of all m roots 
will be the original number.  All of these are multiplicative functions.

(When m is composite, the mth roots can be defined by composing the dth 
and (m/d)th roots for a factor d of m.  Writing the root as 
IntRoot(n,m,k), we have IntRoot(n,m,k) = IntRoot(IntRoot(n, d, 
mod(k,d)), m/d, floor(k/d)).  This reflects the identity floor((e+k)/m) 
= floor((floor((e + mod(k,d)) / d)  + floor(k/d)) / (m/d)).)

Franklin T. Adams-Watters


-----Original Message-----
From: franktaw at netscape.net

For reference, the a's are A000188, and the d's are A007913. A007913 
calls this the square-free part; I don't know how standard this is. 
A000188 doesn't really give a name, just various descriptions. 
Square-free part is perhaps too easily confused with square-free 
kernel, A007947, which is lcm(a,d). 
 
Personally, I think of a as the inner square root, and the product a*d 
(A019544) as the outer square root. 
 
Integer part for a is clearly not acceptable; this is a synonym for the 
floor function. While floor is a better name than integer part for this 
function, this does not make integer part available as a name for 
something else. 
 
Franklin T. Adams-Watters 
 
-----Original Message----- 
From: jrbibers at indiana.edu 
 
Each positive integer has a square root uniquely expressed a product 
a*sqrt(d), where a and d are positive integers and d is squarefree.  
  
What's the standard terminology for the parts a and d of some sqrt(n)?  
  
Some options for a: integer part, ..?  
  
Some options for d: quadratic part, radicand, squarefree part, radical 
part, ..?  
  
Anyhow, the "a-part" and "d-part" of the square root of many 
fundamental integer sequences are absent from EIS.  

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