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a(n) is the number of 2 X 2 Euclid-reduced matrices having coprime elements and determinant n.
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%I #14 Sep 21 2022 12:02:09

%S 1,2,3,4,5,8,7,9,9,14,11,16,13,20,18,19,17,28,19,26,26,32,23,36,25,38,

%T 31,38,29,54,31,41,42,50,38,56,37,56,50,56,41,76,43,62,58,68,47,78,49,

%U 78,66,74,53,92,62,76,74,86,59,114,61,92,78,85,74,124,67,98,90,118

%N a(n) is the number of 2 X 2 Euclid-reduced matrices having coprime elements and determinant n.

%C See Bacher link for the definition of Euclid-reduced.

%H Roland Bacher, <a href="https://arxiv.org/abs/2209.09529">Euclid meets Popeye: The Euclidean Algorithm for 2X2 matrices</a>, arXiv:2209.09529 [math.NT], 2022.

%F a(n) = Sum_{d^2|n} moebius(d)*A357259(n/d^2).

%t f[n_] := DivisorSum[n, # + 1 - n/# &, #^2 >= n &]; a[n_] := DivisorSum[n, MoebiusMu[Sqrt[#]] * f[n/#] &, IntegerQ[Sqrt[#]] &]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2022 *)

%o (PARI) f(n) = sumdiv(n, d, if (d^2 >= n, d + 1 -n/d)); \\ A357259

%o a(n) = sumdiv(n, d, if (issquare(d), moebius(sqrtint(d))*f(n/d)));

%Y Cf. A357259.

%K nonn

%O 1,2

%A _Michel Marcus_, Sep 21 2022