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Search: id:A000252
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| A000252 |
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Number of invertible 2 X 2 matrices mod n. |
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+0
21
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| 1, 6, 48, 96, 480, 288, 2016, 1536, 3888, 2880, 13200, 4608, 26208, 12096, 23040, 24576, 78336, 23328, 123120, 46080, 96768, 79200, 267168, 73728, 300000, 157248, 314928, 193536, 682080, 138240, 892800, 393216, 633600, 470016, 967680, 373248
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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For a prime p, a(p) = (p^2 - 1)*(p^2 - p) (this is the order of GL(2,p)). More generally a(n) is multiplicative: if the canonical factorization of n is the product of p^e(p) over primes p then a(n) = product ((p^(2*e(p)) - p^(2*e(p) - 2)) * (p^(2*e(p)) - p^(2*e(p) - 1))). - Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Apr 05 2001, Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher
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FORMULA
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a(n) = n^4 * product (1-1/p^2)*(1-1/p) = n^4 * product p^(-3)(p^2 - 1)*(p - 1) where the product is over all the primes p that divide n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 18 2001
Multiplicative with a(p^e) = (p-1)^2*(p+1)*p^(4e-3). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
a(n) = A000056(n)*phi(n), where phi is Euler totient function (cf. A000010). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 30 2001
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CROSSREFS
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The number of 2 X 2 matrices mod n with determinant 1 is A000056. The order of GL_2(K) for a finite field K is in sequence A059238.
Cf. A011785, A064767.
Sequence in context: A015553 A071878 A104256 this_sequence A078237 A052651 A153796
Adjacent sequences: A000249 A000250 A000251 this_sequence A000253 A000254 A000255
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jul 21, 2001
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