en>Lbertolotti: /* Linear transformation */

2014-10-04T20:40:36Z

Linear transformation

en>BeyondNormality: /* Related distributions */

2013-12-10T03:45:31Z

Related distributions

← Older revision		Revision as of 05:45, 10 December 2013
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	[http://~~Adosphere~~.com/~~member~~/~~28512~~/~~blog~~/~~view~~/~~66112~~/ ~~home std test~~] ~~Marvella~~ is ~~what you can call her but it~~'s ~~not~~ the ~~most female title out there~~. [http://~~www~~.~~yellowpages~~.~~com~~/~~hialeah-fl~~/~~free-std-testing-centers North Dakota~~] ~~is her birth~~ [http://~~Www~~.~~Pinellashealth~~.com/~~STD~~/~~TypesofSTD~~.~~pdf location~~] ~~but she will have to move one day std home test or another~~. ~~He is truly fond of doing ceramics but he~~ [http://~~chatmast~~.~~com~~/~~index~~.php~~?do=~~/~~BookerSipessk/info~~/ http://~~chatmast~~.~~com~~/~~index~~.php~~?do=~~/~~BookerSipessk/info~~] ~~is having difficulties to find std testing~~ at ~~home time for it. Since she was eighteen she's been operating as a receptionist but her std testing at home promotion never comes.~~<br>~~<br>my blog post:~~ [~~https~~://~~grouponsa~~.~~zendesk~~.~~com~~/~~entries~~/~~96807503-Items-To-Know-While-Confronting-Candida https~~://~~grouponsa~~.~~zendesk~~.~~com~~]		A '''Carol number''' is an integer of the form <math>4^n - 2^{n + 1} - 1</math>. An equivalent formula is <math>(2^n - 1)^2 - 2</math>. The first few Carol numbers are: [[−1 (number)\|−1]], [[7 (number)\|7]], [[47 (number)\|47]], [[223 (number)\|223]], 959, 3967, 16127, 65023, 261119, 1046527 {{OEIS\|id=A093112}}.

			Carol numbers were first studied by Cletus Emmanuel, who named them after a friend, Carol G. Kirnon.<ref>[http://primes.utm.edu/bios/page.php?id=374 Cletus Emmanuel] at [[Prime Pages]]</ref><ref>[http://tech.groups.yahoo.com/group/primenumbers/message/14584 Message to Yahoo primenumbers group] from Cletus Emmanuel</ref>

			For ''n'' > 2, the binary representation of the ''n''-th Carol number is ''n'' − 2 consecutive ones, a single zero in the middle, and ''n'' + 1 more consecutive ones, or to put it algebraically,

			:<math>\sum_{i \ne n + 2}^{2n} 2^{i - 1}.</math>

			So, for example, 47 is 101111 in binary, 223 is 11011111, etc. The difference between the 2''n''-th [[Mersenne prime\|Mersenne number]] and the ''n''-th Carol number is <math>2^{n + 1}</math>. This gives yet another equivalent expression for Carol numbers, <math>(2^{2n} - 1) - 2^{n + 1}</math>. The difference between the ''n''-th [[Kynea number]] and the ''n''-th Carol number is the (''n'' + 2)th [[power of two]].

			Starting with 7, every third Carol number is a multiple of 7. Thus, for a Carol number to also be a [[prime number]], its index ''n'' cannot be of the form 3''x'' + 2 for ''x'' > 0. The first few Carol numbers that are also prime are 7, 47, 223, 3967, 16127 (these are listed in Sloane's {{OEIS2C\|id=A091516}}). {{As of\|2007\|7}}, the largest known Carol number that is also a prime is the Carol number for ''n'' = 253987, which has 152916 digits.<ref>[http://primes.utm.edu/primes/page.php?id=80384 Entry for 253987th Carol number] at [[Prime Pages]]</ref><ref>[http://harvey563.tripod.com/Carol_Kynea.txt Carol Primes and Kynea Primes] by Steven Harvey</ref> It was found by Cletus Emmanuel in May 2007, using the programs MultiSieve and PrimeFormGW. It is the 40th Carol prime.

			The 7th Carol number and 5th Carol prime, 16127, is also a prime when its digits are reversed, so it is the smallest Carol [[emirp]].<ref>[http://primes.utm.edu/curios/page.php/16127.html Prime Curios 16127] at [[Prime Pages]]</ref> The 12th Carol number and 7th Carol prime, 16769023, is also a Carol emirp.<ref>[http://primes.utm.edu/curios/page.php/16769023.html Prime Curios 16769023] at [[Prime Pages]]</ref>

			==References==
			{{reflist}}

			==External links==
			* {{MathWorld\|title=Near-Square Prime\|urlname=Near-SquarePrime}}
			* [http://primes.utm.edu/primes/page.php?id=73109 Prime Database entry for Carol(226749)]
			* [http://primes.utm.edu/primes/page.php?id=77385 Prime Database entry for Carol(248949)]
			[[Category:Integer sequences]]

			{{Prime number classes\|state=collapsed}}
			{{Classes of natural numbers \|state=collapsed}}

			{{numtheory-stub}}

69.127.226.113: Move location parameter to after delta

2012-04-15T02:19:45Z

Move location parameter to after delta

New page

[http://Adosphere.com/member/28512/blog/view/66112/ home std test] Marvella is what you can call her but it's not the most female title out there. [http://www.yellowpages.com/hialeah-fl/free-std-testing-centers North Dakota] is her birth [http://Www.Pinellashealth.com/STD/TypesofSTD.pdf location] but she will have to move one day std home test or another. He is truly fond of doing ceramics but he [http://chatmast.com/index.php?do=/BookerSipessk/info/ http://chatmast.com/index.php?do=/BookerSipessk/info] is having difficulties to find std testing at home time for it. Since she was eighteen she's been operating as a receptionist but her std testing at home promotion never comes.<br><br>my blog post: [https://grouponsa.zendesk.com/entries/96807503-Items-To-Know-While-Confronting-Candida https://grouponsa.zendesk.com]

Generalised hyperbolic distribution - Revision history

en>Lbertolotti: /* Linear transformation */

en>BeyondNormality: /* Related distributions */

69.127.226.113: Move location parameter to after delta