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
Keywords
Links
- 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.
Programs
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Maple
with(numtheory): seq(add(max(d-n/d, 1),d in divisors(n)), n=1..80); # Ridouane Oudra, Oct 30 2023
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Mathematica
a[n_] := DivisorSum[n, # + 1 - n/# &, #^2 >= n &]; Array[a, 100] (* Amiram Eldar, Sep 21 2022 *)
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PARI
a(n) = sumdiv(n, d, if (d^2 >= n, d+1-n/d));
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).
G.f.: Sum_{k>=1} x^(k^2) / (1 - x^k)^2. - Ilya Gutkovskiy, May 17 2024
Comments