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

%I A357259 #30 May 18 2024 09:53:22
%S A357259 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,
%T A357259 38,34,45,29,54,31,52,42,50,38,70,37,56,50,70,41,76,43,73,63,68,47,97,
%U A357259 50,80,66,87,53,100,62,96,74,86,59,132,61,92,85,109,74,124,67,115,90,118
%N A357259 a(n) is the number of 2 X 2 Euclid-reduced matrices having determinant n.
%C A357259 See Bacher link for the definition of Euclid-reduced.
%H A357259 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.
%H A357259 Roland Bacher, <a href="https://doi.org/10.5802/crmath.451">Euclid meets Popeye: The Euclidean Algorithm for 2 X 2 Matrices</a>, Comptes rendus de l’Académie des sciences, Volume 361 (2023), p. 889-895.
%H A357259 MathOverflow, <a href="https://mathoverflow.net/questions/405035/arithmetic-properties-of-positively-reduced-2-times-2-matrices">Arithmetic properties of positively reduced 2×2-matrices</a>, 2021.
%F A357259 a(n) = Sum_{d|n, d^2>=n} d+1-n/d.
%F A357259 From _Ridouane Oudra_, Oct 30 2023: (Start)
%F A357259 a(n) = Sum_{d|n} max(d-n/d, 1).
%F A357259 a(n) = ceiling(tau(n)/2) + (1/2)*Sum_{d|n} abs(d-n/d).
%F A357259 a(n) = A038548(n) + A079667(n). (End)
%F A357259 G.f.: Sum_{k>=1} x^(k^2) / (1 - x^k)^2. - _Ilya Gutkovskiy_, May 17 2024
%p A357259 with(numtheory): seq(add(max(d-n/d, 1),d in divisors(n)), n=1..80); # _Ridouane Oudra_, Oct 30 2023
%t A357259 a[n_] := DivisorSum[n, # + 1 - n/# &, #^2 >= n &]; Array[a, 100] (* _Amiram Eldar_, Sep 21 2022 *)
%o A357259 (PARI) a(n) = sumdiv(n, d, if (d^2 >= n, d+1-n/d));
%Y A357259 Cf. A357260.
%Y A357259 Cf. A038548, A079667.
%K A357259 nonn
%O A357259 1,2
%A A357259 _Michel Marcus_, Sep 21 2022