A357259 a(n) is the number of 2 X 2 Euclid-reduced matrices having determinant n.

1, 2, 3, 5, 5, 8, 7, 11, 10, 14, 11, 19, 13, 20, 18, 24, 17, 30, 19, 31, 26, 32, 23, 44, 26, 38, 34, 45, 29, 54, 31, 52, 42, 50, 38, 70, 37, 56, 50, 70, 41, 76, 43, 73, 63, 68, 47, 97, 50, 80, 66, 87, 53, 100, 62, 96, 74, 86, 59, 132, 61, 92, 85, 109, 74, 124, 67, 115, 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.

Roland Bacher, Euclid meets Popeye: The Euclidean Algorithm for 2 X 2 Matrices, Comptes rendus de l’Académie des sciences, Volume 361 (2023), p. 889-895.

MathOverflow, Arithmetic properties of positively reduced 2×2-matrices, 2021.

Formula

a(n) = Sum_{d|n, d^2>=n} d+1-n/d.

From Ridouane Oudra, Oct 30 2023: (Start)

a(n) = Sum_{d|n} max(d-n/d, 1).

a(n) = ceiling(tau(n)/2) + (1/2)*Sum_{d|n} abs(d-n/d).

a(n) = A038548(n) + A079667(n). (End)

Maple

with(numtheory): seq(add(max(d-n/d, 1), d in divisors(n)), n=1..80); # Ridouane Oudra, Oct 30 2023

Mathematica

a[n_] := DivisorSum[n, # + 1 - n/# &, #^2 >= n &]; Array[a, 100] (* Amiram Eldar, Sep 21 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (d^2 >= n, d+1-n/d));

Crossrefs

Cf. A357260.

Cf. A038548, A079667.

Sequence in context: A267453 A063914 A209187 * A166250 A174088 A304493

Adjacent sequences: A357256 A357257 A357258 * A357260 A357261 A357262

Keyword

nonn

Author

Michel Marcus, Sep 21 2022

Status

approved