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A305186
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Number of invertible 4 X 4 matrices mod n.
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9
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1, 20160, 24261120, 1321205760, 116064000000, 489104179200, 27811094169600, 86586540687360, 1044361663787520, 2339850240000000, 41393302251840000, 32053931488051200, 610296923230525440, 560671658459136000, 2815842631680000000, 5674535530486824960
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OFFSET
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1,2
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COMMENTS
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Order of the group GL(4,Z_n).
Order of the automorphism group of the group (C_n)^4, where C_n is the cyclic group of order n.
For n > 2, a(n) is divisible by 23040.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = (p - 1)*(p^2 - 1)*(p^3 - 1)*(p^4 - 1)*p^(16*e-10).
a(n) = n^16*Product_{primes p dividing n} (1 - 1/p^4)*(1 - 1/p^3)*(1 - 1/p^2)*(1 - 1/p).
Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} ((p^11 - p^9 - p^8 + 2*p^5 - p^2 - p + 1)/p^11) = 0.02958150406... . - Amiram Eldar, Oct 23 2022
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MATHEMATICA
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{1}~Join~Array[#^16*Product[(1 - 1/p^4) (1 - 1/p^3) (1 - 1/p^2) (1 - 1/p), {p, FactorInteger[#][[All, 1]]}] &, 12, 2] (* Michael De Vlieger, May 27 2018 *)
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PROG
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(PARI) a(n)=my(f=factor(n)[, 1]); n^16*prod(i=1, #f, (1-1/f[i]^4)*(1-1/f[i]^3)*(1-1/f[i]^2)*(1-1/f[i]))
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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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STATUS
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approved
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