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en>Ugog Nizdast: spacing

2014-10-04T17:59:47Z

spacing

← Older revision		Revision as of 19:59, 4 October 2014
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	~~In [[number theory]], a '''perfect totient number'''~~ is ~~an [[integer]] that~~ is ~~equal to the sum of its iterated [[totient]]s~~. ~~That~~ is~~, we apply~~ the ~~[[totient function]] to a number ''n'', apply it again to the resulting totient, and so on, until the number 1~~ is ~~reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number~~. ~~Or to put it algebraically, if~~		Hello and welcome. My name is Numbers Wunder. North Dakota is our beginning location. To play baseball is the hobby he will never quit performing. Supervising is my profession.<br><br>My web-site - [http://spermdonorinfo.co.uk/groups/eliminate-a-yeast-infection-with-one-of-these-tips/ home std test]
	:<~~math~~>~~n = \sum_{i = 1}^{c + 1} \varphi^i(n),~~<~~/math~~>
	~~where~~
	~~:<math>\varphi^i(n)=\left\{\begin{matrix}\varphi(n)&\mbox{ if } i=1\\ \varphi(\varphi^{i~~-~~1}(n))&\mbox{ otherwise}\end{matrix}\right.</math>~~
	~~is the iterated totient function and ''c'' is the integer such that~~
	~~:<math>\displaystyle\varphi^c(n)=2,</math>~~
	~~then ''n'' is a perfect totient number.~~

	~~The first few perfect totient numbers are~~
	:[[3 (number)\|3]], [[9 (number)\|9]], [[15 (number)\|15]], [[27 (number)\|27]], [[39 (number)\|39]], [[81 (number)\|81]], [[111 (number)\|111]], [[183 (number)\|183]], [[243 (number)\|243]], [[255 (number)\|255]], [[327 (number)\|327]], [[363 (number)\|363]], 471, [[729 (number)\|729]], 2187, 2199, 3063, 4359, 4375, ... {{OEIS\|id=A082897}}.

	~~For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.~~

	~~== Multiples and powers of three ==~~

	It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
	:~~<math>\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2\times 3^{k-1}.<~~/~~math>~~

	~~Venkataraman (1975) found another family of perfect totient numbers: if ''p'' = 4×3<sup>k<~~/~~sup>+1 is prime, then 3''p'' is a perfect totient number~~. ~~The values of ''k'' leading to perfect totient numbers in this way are~~
	~~:0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635,~~ .~~.. {{OEIS\|id=A005537}}.~~

	More generally if ''p'' is a [[prime number]] greater than 3, and 3''p'' is a perfect totient number, then ''p'' ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all ''p'' of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9''p'' is a perfect totient number then ''p'' is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3<sup>k</~~sup>''p'' where ''p'' is prime and ''k'' > 3.~~

	~~==References==~~
	*{{cite journal
	~~\| author = Pérez~~-~~Cacho Villaverde, Laureano~~
	~~\| title = Sobre la suma de indicadores de ordenes sucesivos~~
	~~\| journal = Revista Matematica Hispano~~-~~Americana~~
	~~\| volume = 5~~
	~~\| issue = 3~~
	~~\| year = 1939~~
	~~\| pages = 45–50}}~~

	*{{cite book
	~~\| author = Guy, Richard K.~~
	~~\| authorlink = Richard K. Guy~~
	~~\| title = Unsolved Problems in Number Theory~~
	~~\| location = New York~~
	~~\| publisher = Springer~~-~~Verlag~~
	~~\| year = 2004~~
	~~\| isbn=0~~-~~387~~-~~20860~~-7
	~~\| page = §B41}}~~

	*{{cite journal
	~~\| author = Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L.~~
	~~\| title = On perfect totient numbers~~
	~~\| journal = [[Journal~~ of ~~Integer Sequences]]~~
	~~\| volume = 6~~
	~~\| year = 2003~~
	~~\| issue = 4~~
	~~\| pages = 03.4.5~~
	~~\| id = {{MathSciNet \| id = 2051959}}~~
	~~\| url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf}}~~

	*{{cite journal
	~~\| author = Luca, Florian~~
	~~\| title = On the distribution of perfect totients~~
	~~\| journal = Journal of Integer Sequences~~
	~~\| volume = 9~~
	~~\| year = 2006~~
	~~\| issue = 4~~
	~~\| pages = 06.4.4~~
	~~\| id = {{MathSciNet \| id = 2247943}}~~
	~~\| url = http:~~/~~/www.emis.ams.org/journals/JIS/VOL9/Luca/luca66.pdf}}~~

	*{{cite conference
	~~\| author = Mohan, A. L.; Suryanarayana, D.~~
	~~\| title = Perfect totient numbers~~
	~~\| booktitle = Number theory (Mysore, 1981)~~
	~~\| pages = 101–105~~
	~~\| publisher = Lecture Notes in Mathematics, vol. 938, Springer-Verlag~~
	~~\| year = 1982~~
	~~\| id = {{MathSciNet \| id = 0665442}}}}~~

	*{{cite journal
	~~\| author = Venkataraman, T.~~
	~~\| title = Perfect totient number~~
	~~\| journal = [[The Mathematics Student]]~~
	~~\| volume = 43~~
	~~\| year = 1975~~
	~~\| page = 178~~
	~~\| id = {{MathSciNet \| id = 0447089}}}}~~

	~~{{PlanetMath attribution\|id=8741\|title=Perfect Totient Number}}~~

	~~{{Classes of natural numbers}}~~
	~~[[Category:Integer sequences]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q4381560

2013-03-16T14:09:41Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q4381560

← Older revision		Revision as of 16:09, 16 March 2013
Line 1:		Line 1:
	~~Golda~~ is ~~what~~'~~s created~~ on ~~my birth certificate even though~~ it is not the ~~name on my beginning certification~~. ~~To play domino~~ is ~~some thing I truly appreciate doing~~. ~~For years he~~'~~s been living~~ in ~~Alaska~~ and ~~he doesn~~'~~t plan on changing it~~. ~~Office supervising~~ is ~~what she does for~~ a ~~residing~~.<br><br>~~My web~~ page: [http://~~chungmuroresidence~~.~~com~~/xe/~~reservation_branch2~~/~~152663 best psychic readings~~]		In [[number theory]], a '''perfect totient number''' is an [[integer]] that is equal to the sum of its iterated [[totient]]s. That is, we apply the [[totient function]] to a number ''n'', apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number. Or to put it algebraically, if
			:<math>n = \sum_{i = 1}^{c + 1} \varphi^i(n),</math>
			where
			:<math>\varphi^i(n)=\left\{\begin{matrix}\varphi(n)&\mbox{ if } i=1\\ \varphi(\varphi^{i-1}(n))&\mbox{ otherwise}\end{matrix}\right.</math>
			is the iterated totient function and ''c'' is the integer such that
			:<math>\displaystyle\varphi^c(n)=2,</math>
			then ''n'' is a perfect totient number.

			The first few perfect totient numbers are
			:[[3 (number)\|3]], [[9 (number)\|9]], [[15 (number)\|15]], [[27 (number)\|27]], [[39 (number)\|39]], [[81 (number)\|81]], [[111 (number)\|111]], [[183 (number)\|183]], [[243 (number)\|243]], [[255 (number)\|255]], [[327 (number)\|327]], [[363 (number)\|363]], 471, [[729 (number)\|729]], 2187, 2199, 3063, 4359, 4375, ... {{OEIS\|id=A082897}}.

			For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.

			== Multiples and powers of three ==

			It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
			:<math>\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2\times 3^{k-1}.</math>

			Venkataraman (1975) found another family of perfect totient numbers: if ''p'' = 4×3<sup>k</sup>+1 is prime, then 3''p'' is a perfect totient number. The values of ''k'' leading to perfect totient numbers in this way are
			:0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... {{OEIS\|id=A005537}}.

			More generally if ''p'' is a [[prime number]] greater than 3, and 3''p'' is a perfect totient number, then ''p'' ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all ''p'' of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9''p'' is a perfect totient number then ''p'' is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3<sup>k</sup>''p'' where ''p'' is prime and ''k'' > 3.

			==References==
			*{{cite journal
			\| author = Pérez-Cacho Villaverde, Laureano
			\| title = Sobre la suma de indicadores de ordenes sucesivos
			\| journal = Revista Matematica Hispano-Americana
			\| volume = 5
			\| issue = 3
			\| year = 1939
			\| pages = 45–50}}

			*{{cite book
			\| author = Guy, Richard K.
			\| authorlink = Richard K. Guy
			\| title = Unsolved Problems in Number Theory
			\| location = New York
			\| publisher = Springer-Verlag
			\| year = 2004
			\| isbn=0-387-20860-7
			\| page = §B41}}

			*{{cite journal
			\| author = Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L.
			\| title = On perfect totient numbers
			\| journal = [[Journal of Integer Sequences]]
			\| volume = 6
			\| year = 2003
			\| issue = 4
			\| pages = 03.4.5
			\| id = {{MathSciNet \| id = 2051959}}
			\| url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf}}

			*{{cite journal
			\| author = Luca, Florian
			\| title = On the distribution of perfect totients
			\| journal = Journal of Integer Sequences
			\| volume = 9
			\| year = 2006
			\| issue = 4
			\| pages = 06.4.4
			\| id = {{MathSciNet \| id = 2247943}}
			\| url = http://www.emis.ams.org/journals/JIS/VOL9/Luca/luca66.pdf}}

			*{{cite conference
			\| author = Mohan, A. L.; Suryanarayana, D.
			\| title = Perfect totient numbers
			\| booktitle = Number theory (Mysore, 1981)
			\| pages = 101–105
			\| publisher = Lecture Notes in Mathematics, vol. 938, Springer-Verlag
			\| year = 1982
			\| id = {{MathSciNet \| id = 0665442}}}}

			*{{cite journal
			\| author = Venkataraman, T.
			\| title = Perfect totient number
			\| journal = [[The Mathematics Student]]
			\| volume = 43
			\| year = 1975
			\| page = 178
			\| id = {{MathSciNet \| id = 0447089}}}}

			{{PlanetMath attribution\|id=8741\|title=Perfect Totient Number}}

			{{Classes of natural numbers}}
			[[Category:Integer sequences]]

en>Arthur Rubin: Reverted 8 edits by 193.62.7.228 (talk): Not. The only reason I didn't tag it as vandalism, is that the poster may not know it's nonsense. (TW)

2012-05-22T17:12:07Z

Reverted 8 edits by 193.62.7.228 (talk): Not. The only reason I didn't tag it as vandalism, is that the poster may not know it's nonsense. (TW)

New page

Golda is what's created on my birth certificate even though it is not the name on my beginning certification. To play domino is some thing I truly appreciate doing. For years he's been living in Alaska and he doesn't plan on changing it. Office supervising is what she does for a residing.<br><br>My web page: [http://chungmuroresidence.com/xe/reservation_branch2/152663 best psychic readings]

← Older revision		Revision as of 19:59, 4 October 2014
Line 1:		Line 1:
	~~In [[number theory]], a '''perfect totient number'''~~ is ~~an [[integer]] that~~ is ~~equal to the sum of its iterated [[totient]]s~~. ~~That~~ is~~, we apply~~ the ~~[[totient function]] to a number ''n'', apply it again to the resulting totient, and so on, until the number 1~~ is ~~reached, and add together the resulting sequence of numbers; if the sum equals ''n'', then ''n'' is a perfect totient number~~. ~~Or to put it algebraically, if~~		Hello and welcome. My name is Numbers Wunder. North Dakota is our beginning location. To play baseball is the hobby he will never quit performing. Supervising is my profession.<br><br>My web-site - [http://spermdonorinfo.co.uk/groups/eliminate-a-yeast-infection-with-one-of-these-tips/ home std test]
	:<~~math~~>~~n = \sum_{i = 1}^{c + 1} \varphi^i(n),~~<~~/math~~>
	~~where~~
	~~:<math>\varphi^i(n)=\left\{\begin{matrix}\varphi(n)&\mbox{ if } i=1\\ \varphi(\varphi^{i~~-~~1}(n))&\mbox{ otherwise}\end{matrix}\right.</math>~~
	~~is the iterated totient function and ''c'' is the integer such that~~
	~~:<math>\displaystyle\varphi^c(n)=2,</math>~~
	~~then ''n'' is a perfect totient number.~~

	~~The first few perfect totient numbers are~~
	:[[3 (number)\|3]], [[9 (number)\|9]], [[15 (number)\|15]], [[27 (number)\|27]], [[39 (number)\|39]], [[81 (number)\|81]], [[111 (number)\|111]], [[183 (number)\|183]], [[243 (number)\|243]], [[255 (number)\|255]], [[327 (number)\|327]], [[363 (number)\|363]], 471, [[729 (number)\|729]], 2187, 2199, 3063, 4359, 4375, ... {{OEIS\|id=A082897}}.

	~~For example, start with 327. Then φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.~~

	~~== Multiples and powers of three ==~~

	It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
	:~~<math>\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2\times 3^{k-1}.<~~/~~math>~~

	~~Venkataraman (1975) found another family of perfect totient numbers: if ''p'' = 4×3<sup>k<~~/~~sup>+1 is prime, then 3''p'' is a perfect totient number~~. ~~The values of ''k'' leading to perfect totient numbers in this way are~~
	~~:0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635,~~ .~~.. {{OEIS\|id=A005537}}.~~

	More generally if ''p'' is a [[prime number]] greater than 3, and 3''p'' is a perfect totient number, then ''p'' ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all ''p'' of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9''p'' is a perfect totient number then ''p'' is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3<sup>k</~~sup>''p'' where ''p'' is prime and ''k'' > 3.~~

	~~==References==~~
	*{{cite journal
	~~\| author = Pérez~~-~~Cacho Villaverde, Laureano~~
	~~\| title = Sobre la suma de indicadores de ordenes sucesivos~~
	~~\| journal = Revista Matematica Hispano~~-~~Americana~~
	~~\| volume = 5~~
	~~\| issue = 3~~
	~~\| year = 1939~~
	~~\| pages = 45–50}}~~

	*{{cite book
	~~\| author = Guy, Richard K.~~
	~~\| authorlink = Richard K. Guy~~
	~~\| title = Unsolved Problems in Number Theory~~
	~~\| location = New York~~
	~~\| publisher = Springer~~-~~Verlag~~
	~~\| year = 2004~~
	~~\| isbn=0~~-~~387~~-~~20860~~-7
	~~\| page = §B41}}~~

	*{{cite journal
	~~\| author = Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L.~~
	~~\| title = On perfect totient numbers~~
	~~\| journal = [[Journal~~ of ~~Integer Sequences]]~~
	~~\| volume = 6~~
	~~\| year = 2003~~
	~~\| issue = 4~~
	~~\| pages = 03.4.5~~
	~~\| id = {{MathSciNet \| id = 2051959}}~~
	~~\| url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf}}~~

	*{{cite journal
	~~\| author = Luca, Florian~~
	~~\| title = On the distribution of perfect totients~~
	~~\| journal = Journal of Integer Sequences~~
	~~\| volume = 9~~
	~~\| year = 2006~~
	~~\| issue = 4~~
	~~\| pages = 06.4.4~~
	~~\| id = {{MathSciNet \| id = 2247943}}~~
	~~\| url = http:~~/~~/www.emis.ams.org/journals/JIS/VOL9/Luca/luca66.pdf}}~~

	*{{cite conference
	~~\| author = Mohan, A. L.; Suryanarayana, D.~~
	~~\| title = Perfect totient numbers~~
	~~\| booktitle = Number theory (Mysore, 1981)~~
	~~\| pages = 101–105~~
	~~\| publisher = Lecture Notes in Mathematics, vol. 938, Springer-Verlag~~
	~~\| year = 1982~~
	~~\| id = {{MathSciNet \| id = 0665442}}}}~~

	*{{cite journal
	~~\| author = Venkataraman, T.~~
	~~\| title = Perfect totient number~~
	~~\| journal = [[The Mathematics Student]]~~
	~~\| volume = 43~~
	~~\| year = 1975~~
	~~\| page = 178~~
	~~\| id = {{MathSciNet \| id = 0447089}}}}~~

	~~{{PlanetMath attribution\|id=8741\|title=Perfect Totient Number}}~~

	~~{{Classes of natural numbers}}~~
	~~[[Category:Integer sequences]~~]