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%I #23 Feb 13 2024 02:19:33
%S 1,8,48,128,480,384,2016,2048,3888,3840,13200,6144,26208,16128,23040,
%T 32768,78336,31104,123120,61440,96768,105600,267168,98304,300000,
%U 209664,314928,258048,682080,184320,892800,524288,633600,626688,967680,497664,1822176
%N Number of the Lipschitz quaternions in a reduced system modulo n.
%H Amiram Eldar, <a href="/A227499/b227499.txt">Table of n, a(n) for n = 1..10000</a>
%H Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, <a href="http://dx.doi.org/10.5486/PMD.2015.7098">Counting invertible sums of squares modulo n and a new generalization of Euler's totient function</a>, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; <a href="https://arxiv.org/abs/1403.7878">arXiv preprint</a>, arXiv:1403.7878 [math.NT], 2014.
%F Multiplicative: a(2^s) = 2^(4s-1); a(3^s) = 16*3^(4s-3); a(5^s) = 32*3*5^(4s-3).
%F From _Amiram Eldar_, Feb 13 2024: (Start)
%F 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.
%F 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)).
%F 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).
%F 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)
%t 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]
%t 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 *)
%o (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
%Y Cf. A079458, A060968, A087784.
%K nonn,easy,mult
%O 1,2
%A _José María Grau Ribas_, Jul 13 2013
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