This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A304182 #25 Dec 31 2022 03:36:22
%S A304182 1,3,2,4,2,6,2,4,2,6,2,8,2,6,4,4,2,6,2,8,4,6,2,8,2,6,2,8,2,12,2,4,4,6,
%T A304182 4,8,2,6,4,8,2,12,2,8,4,6,2,8,2,6,4,8,2,6,4,8,4,6,2,16,2,6,4,4,4,12,2,
%U A304182 8,4,12,2,8,2,6,4,8,4,12,2,8,2,6,2,16,4
%N A304182 Number of primitive inequivalent mirror-symmetric sublattices of rectangular lattice of index n.
%H A304182 Álvar Ibeas, <a href="/A304182/b304182.txt">Table of n, a(n) for n = 1..10000</a>
%H A304182 John S. Rutherford, <a href="https://doi.org/10.1107/S010876730804333X">Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type</a>, Acta Cryst. (2009). A65, 156-163. [See Table 4. Contains errors for n = 24 and 28.]
%F A304182 From _Álvar Ibeas_, Mar 18 2021: (Start)
%F A304182 For n odd, a(n) = A034444(n) = 2^(A001221(n)).
%F A304182 For n even, a(n) = A034444(n) + A034444(n/2). If 4|n, a(n) = 2^(A001221(n) + 1); otherwise, a(n) = 3 * 2^(A001221(n) - 1).
%F A304182 Multiplicative with a(2) = 3, a(2^e) = 4 (for e>1), and a(p^e) = 2 (for p>2).
%F A304182 Dirichlet g.f.: (1+2^(-s)) * zeta(s)^2 / zeta(2s).
%F A304182 (End)
%F A304182 Sum_{k=1..n} a(k) ~ (log(n) + 2*gamma - log(2)/3 - 2*zeta'(2)/zeta(2) - 1)*9*n/Pi^2, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 31 2022
%e A304182 There are 6 = A001615(4) lattices in Z^2 whose quotient group is C_4. The reflection through an axis relates <(4,0), (1,1)> and <(4,0), (3,1)>. The remaining 4 = a(4) lattices are fixed.
%t A304182 f[p_, e_] := If[p == 2, If[e == 1, 3, 4], 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 22 2022 *)
%Y A304182 Cf. A069735 (not only primitive sublattices), A304183 (primitive oblique sublattices), A069734 (all sublattices).
%Y A304182 Cf. other columns of tables 4 and 5 from [Rutherford, 2009]: A001615, A060594, A157223, A000089, A157224, A000086, A157227, A019590, A157228, A157226, A157230, A157231, A154272, A157235.
%Y A304182 Cf. A034444, A001221, A001620, A306016.
%K A304182 nonn,mult
%O A304182 1,2
%A A304182 _Andrey Zabolotskiy_, May 07 2018