en>Cydebot: Robot - Removing category Iterated binary operations per CFD at Wikipedia:Categories for discussion/Log/2013 June 23.

2013-07-25T02:21:24Z

Robot - Removing category Iterated binary operations per CFD at Wikipedia:Categories for discussion/Log/2013 June 23.

← Older revision		Revision as of 04:21, 25 July 2013
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	The ~~name~~ of the ~~writer~~ is ~~Jayson~~. ~~What me~~ and ~~my family adore~~ is ~~to climb but I~~'~~m considering on beginning some thing new~~. ~~Ohio~~ is ~~exactly~~ where ~~her home~~ is. ~~He works as a bookkeeper~~.<br><br>my page :~~: online psychics~~ ([http://~~www~~.~~skullrocker~~.~~com~~/~~blogs~~/~~post/10991 Highly recommended Internet~~ page])		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—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—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—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—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—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'')
			\|  {{OEIS2C\|A034448}} is σ<sub>1</sub>(''n'')
			\|  {{OEIS2C\|A034676}} to {{OEIS2C\|A034682}} are σ<sub>2</sub>(''n'') to σ<sub>8</sub>(''n'')
			\|  {{OEIS2C\|A068068}} is σ<sup>(o)*</sup><sub>0</sub>(''n'')
			\|  {{OEIS2C\|A192066}} is σ<sup>(o)*</sup><sub>1</sub>(''n'')
			}}

			{{Divisor classes}}

			[[Category:Number theory]]

en>R'n'B: Fix links to disambiguation page Infinite

2012-08-22T23:59:03Z

Fix links to disambiguation page Infinite

New page

The name of the writer is Jayson. What me and my family adore is to climb but I'm considering on beginning some thing new. Ohio is exactly where her home is. He works as a bookkeeper.<br><br>my page :: online psychics ([http://www.skullrocker.com/blogs/post/10991 Highly recommended Internet page])

Iterated binary operation - Revision history

en>Cydebot: Robot - Removing category Iterated binary operations per CFD at Wikipedia:Categories for discussion/Log/2013 June 23.

en>R'n'B: Fix links to disambiguation page Infinite