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A227499
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Number of the Lipschitz quaternions in a reduced system modulo n.
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16
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1, 8, 48, 128, 480, 384, 2016, 2048, 3888, 3840, 13200, 6144, 26208, 16128, 23040, 32768, 78336, 31104, 123120, 61440, 96768, 105600, 267168, 98304, 300000, 209664, 314928, 258048, 682080, 184320, 892800, 524288, 633600, 626688, 967680, 497664, 1822176
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OFFSET
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1,2
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LINKS
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FORMULA
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Multiplicative: a(2^s) = 2^(4s-1); a(3^s) = 16*3^(4s-3); a(5^s) = 32*3*5^(4s-3).
Multiplicative with a(2^e) = 2^(4*e-1), and a(p^e) = p^(4*e-3) * (p-1)^2 * (p+1) for an odd prime p.
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^(s-3)) * Product_{p prime > 2} (1 - 1/p^(s-3) - (p-1)/p^(s-1)).
Sum_{k=1..n} a(k) = (12/55) * c * n^5 + O(n^4 * log(n)), where c = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.53589615382833799980... (Calderón et al., 2015).
Sum_{n>=1} 1/a(n) = (17*Pi^8/57240) * Product_{p prime} (1 - 2/p^2 + 1/p^4 + 1/p^5 + 2/p^6 - 1/p^8) = 1.16039588611967540703... . (End)
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MATHEMATICA
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cuater[n_] := Flatten[Table[{a, b, c, d}, {a, n}, {b, n}, {c, n}, {d, n}], 3]; a[n_] := Length@Select[cuater[n], GCD[#.#, n] == 1 &]; Array[a, 20]
f[p_, e_] := (p-1)*p^(4*e-1) * If[p == 2, 1, 1 - 1/p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 13 2024 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p-1)*p^(4*e-1) * if(p == 2, 1, 1 - 1/p^2)); } \\ Amiram Eldar, Feb 13 2024
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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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