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A064767
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Order of automorphism group of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
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14
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1, 168, 11232, 86016, 1488000, 1886976, 33784128, 44040192, 221079456, 249984000, 2124276000, 966131712, 9726417792, 5675733504, 16713216000, 22548578304, 111203278848, 37141348608, 304812862560, 127991808000
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OFFSET
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1,2
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COMMENTS
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Also number of 3 X 3 invertible matrices over the ring Z/nZ. - Max Alekseyev, Nov 02 2007
Order of the group GL(3,Z_n). For n > 2, a(n) is divisible by 96. - Jianing Song, Nov 24 2018
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LINKS
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FORMULA
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a(n) = n^9*Product_{primes p dividing n} ((1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)). This also gives a formula for A011785.
Multiplicative with a(p^e) = p^(9*e-6)*(p^3 - 1)*(p^2 - 1)*(p - 1). - Vladeta Jovovic, Nov 18 2001
Sum_{k=1..n} a(k) ~ c * n^10, where c = (1/10) * Product_{p prime} ((p^7 - p^5 - p^4 + p^2 + p - 1)/p^7) = 0.05123382571... . - Amiram Eldar, Oct 23 2022
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MATHEMATICA
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a[n_] := n^9*Times @@ Function[p, (1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p)] /@ FactorInteger[n][[All, 1]]; a[1] = 1; Array[a, 20] (* Jean-François Alcover, Mar 21 2017 *)
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PROG
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(PARI) a(n) = n^9*prod(k=2, n, if (!isprime(k) || (n % k), 1, (1-1/k^3)*(1-1/k^2)*(1-1/k))); \\ Michel Marcus, Jun 30 2015
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 24 2001
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EXTENSIONS
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STATUS
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approved
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