Iterated binary operation

In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and $\frac{b}{a}$ are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and $\frac{60}{5}=12$ have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and $\frac{60}{6}=10$ have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.

Equivalently, a given divisor a of b is a unitary divisor iff every prime factor of a has the same multiplicity in a as it has in b.

The sum of unitary divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*_k(n):

${\sigma }_k^{*}(n)=\sum _{\genfrac{}{}{0ex}{}{d\mid n}{\gcd (d,n/d)=1}}{d}^k.$

If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.

Properties

The number of unitary divisors of a number n is 2^k, where k is the number of distinct prime factors of n. The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise.

Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is

$\frac{\zeta (s)\zeta (s-k)}{\zeta (2s-k)}=\sum _{n\ge 1}\frac{{\sigma }_k^{*}(n)}{n^s}.$

Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is

${\sigma }_k^{(o)*}(n)=\sum _{\genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{d\mid n}{d\equiv 1(\mathrm{mod}2)}}{\gcd (d,n/d)=1}}{d}^k.$

It is also multiplicative, with Dirichlet generating function

$\frac{\zeta (s)\zeta (s-k)(1-2^{k-s})}{\zeta (2s-k)(1-2^{k-2s})}=\sum _{n\ge 1}\frac{{\sigma }_k^{(o)*}(n)}{n^s}.$

Bi-unitary divisors

A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. The number of bi-unitary divisors of n is a multiplicative function of n with average order $A\log x$ where

$A=\prod _p\left(1-\frac{p-1}{p^2(p+1)}\right).$

A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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