Iterated binary operation: Difference between revisions

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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 <math>\frac{b}{a}</math> are [[coprime]], having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and <math>\frac{60}{5}=12</math> have only 1 as a common factor, while 6 is a [[divisor]] but not a unitary divisor of 60, as 6 and <math>\frac{60}{6}=10</math> 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 (mathematics)|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*<sub>k</sub>(''n''):
 
:<math>\sigma_k^*(n) = \sum_{d\mid n \atop \gcd(d,n/d)=1} \!\! d^k.</math>
 
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<sup>''k''</sup>, where ''k'' is the number of distinct [[prime factor]]s 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 function]]s of ''n'' that are not completely multiplicative. The [[Dirichlet generating function]] is
 
:<math>\frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} = \sum_{n\ge 1}\frac{\sigma_k^*(n)}{n^s}.</math>
 
== Odd unitary divisors ==
The sum of the ''k''-th powers of the odd unitary divisors is
 
:<math>\sigma_k^{(o)*}(n) = \sum_{{d\mid n \atop d\equiv 1 \pmod 2} \atop \gcd(d,n/d)=1} \!\! d^k.</math>
 
It is also multiplicative, with Dirichlet generating function
 
:<math>\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}.</math>
 
==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 of an arithmetic function|average order]] <math>A \log x</math> where<ref name=Ivic395>Ivić (1985) p.395</ref>
 
:<math>A = \prod_p\left({1 - \frac{p-1}{p^2(p+1)} }\right) \ . </math>
 
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.<ref name=HNTI115>Sandor et al (2006) p.115</ref>
 
==References==
{{reflist}}
* {{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=Unsolved Problems in Number Theory|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7 | page=84}}  Section B3.
* {{cite book | title=My Numbers, My Friends: Popular Lectures on Number Theory | author=Paulo Ribenboim | authorlink=Paulo Ribenboim | publisher=Springer-Verlag | year=2000 | isbn=0-387-98911-0 | page=352 }}
* {{cite news |first1=Eckford|last1=Cohen|title=A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion|journal=Pacific J. Math.|volume=9|number=1
|pages=13&mdash;23|year=1959|mr=0109806}}
* {{cite news|first1=Eckford|last1=Cohen|title=Arithmetical functions associated with the unitary divisors of an integer|journal=[[Mathematische Zeitschrift]]|volume=74|year=1960|pages=66&mdash;80|mr=0112861
|doi=10.1007/BF01180473}}
* {{cite news|first1=Eckford|last1=Cohen|title=The number of unitary divisors of an integer|volume=67|number=9|pages=879&mdash;880|mr=0122790|year=1960|journal=[[American mathematical monthly]]
}}
* {{cite news|first1=Graeme L.|last1=Cohen|title=On an integers' infinitary divisors|volume=54|number=189
|pages=395&mdash;411|mr=0993927|doi=10.1090/S0025-5718-1990-0993927-5|journal=Math. Comp.|year=1990}}
* {{cite news|first1=Graeme L.|last1=Cohen|title=Arithmetic functions associated with infinitary divisors of an integer|volume=16|number=2|pages=373&mdash;383|doi=10.1155/S0161171293000456|journal=Intl. J. Math. Math. Sci.|year=1993}}
* {{cite web|first1=Steven|last1=Finch|title=Unitarism and Infinitarism|url=http://algo.inria.fr/csolve/try.pdf|year=2004}}
* {{cite book | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | page=395 }}
* {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}
 
== External links ==
* {{MathWorld |urlname=UnitaryDivisor |title=Unitary Divisor}}
 
===[[OEIS]] sequences===
{{hlist
| {{OEIS2C|A034444}} is σ<sub>0</sub>(''n'')&nbsp;
| &nbsp;{{OEIS2C|A034448}} is σ<sub>1</sub>(''n'')&nbsp;
| &nbsp;{{OEIS2C|A034676}} to {{OEIS2C|A034682}} are σ<sub>2</sub>(''n'') to σ<sub>8</sub>(''n'')&nbsp;
| &nbsp;{{OEIS2C|A068068}} is σ<sup>(o)*</sup><sub>0</sub>(''n'')&nbsp;
| &nbsp;{{OEIS2C|A192066}} is σ<sup>(o)*</sup><sub>1</sub>(''n'')&nbsp;
}}
 
{{Divisor classes}}
 
[[Category:Number theory]]

Latest revision as of 03:21, 25 July 2013

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 ba are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 605=12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 606=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):

σk*(n)=∑d∣ngcd⁡(d,n/d)=1dk.

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 2k, 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

ζ(s)ζ(s−k)ζ(2s−k)=∑n≥1σk*(n)ns.

Odd unitary divisors

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

σk(o)*(n)=∑d∣nd≡1(mod2)gcd⁡(d,n/d)=1dk.

It is also multiplicative, with Dirichlet generating function

ζ(s)ζ(s−k)(1−2k−s)ζ(2s−k)(1−2k−2s)=∑n≥1σk(o)*(n)ns.

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 Alog⁡x where[1]

A=∏p(1−p−1p2(p+1)).

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.[2]

References

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External links



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OEIS sequences

Template:Hlist

Template:Divisor classes

  1. ↑ Ivić (1985) p.395
  2. ↑ Sandor et al (2006) p.115
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