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

1,2

Comments

See Bacher link for the definition of Euclid-reduced.

Links

Table of n, a(n) for n=1..70.

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

Formula

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

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 *)

Prog

(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)));

Crossrefs

Cf. A357259.

Sequence in context: A319605 A352047 A245822 * A069797 A158979 A233249

Adjacent sequences: A357257 A357258 A357259 * A357261 A357262 A357263

Keyword

nonn

Author

Michel Marcus, Sep 21 2022

Status

approved