A357260 a(n) is the number of 2 X 2 Euclid-reduced matrices having coprime elements and determinant n.

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, 31, 38, 29, 54, 31, 41, 42, 50, 38, 56, 37, 56, 50, 56, 41, 76, 43, 62, 58, 68, 47, 78, 49, 78, 66, 74, 53, 92, 62, 76, 74, 86, 59, 114, 61, 92, 78, 85, 74, 124, 67, 98, 90, 118

Offset: 1

Author

Michel Marcus, Sep 21 2022

Keywords

• nonn

Comments

See Bacher link for the definition of Euclid-reduced.

Links

• Roland Bacher, Euclid meets Popeye: The Euclidean Algorithm for 2X2 matrices, arXiv:2209.09529 [math.NT], 2022.

Crossrefs

Cf. A357259.

Programs

Mathematica

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 *)

PARI

f(n) = sumdiv(n, d, if (d^2 >= n, d + 1 -n/d)); \\ A357259
a(n) = sumdiv(n, d, if (issquare(d), moebius(sqrtint(d))*f(n/d)));

Formula

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