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 A045771 #35 May 27 2025 01:09:15
%S A045771 1,1,8,1,12,8,16,1,41,12,24,8,28,16,96,1,36,41,40,12,128,24,48,8,97,
%T A045771 28,176,16,60,96,64,1,192,36,192,41,76,40,224,12,84,128,88,24,492,48,
%U A045771 96,8,177,97,288,28,108,176,288,16,320,60,120,96,124,64,656,1
%N A045771 Number of similar sublattices of index n^2 in root lattice D_4.
%H A045771 Michael De Vlieger, <a href="/A045771/b045771.txt">Table of n, a(n) for n = 1..16384</a>
%H A045771 Michael Baake and Robert V. Moody, <a href="http://dx.doi.org/10.4153/CJM-1999-057-0">Similarity submodules and root systems in four dimensions</a>, Canad. J. Math., Vol. 51, No. 6 (1999), 1258-1276.
%H A045771 Michael Baake and Peter Zeiner, "Similar Sublattices", Ch. 3.5 in <a href="https://doi.org/10.1017/9781139033862">Aperiodic Order</a>, Vol. 2: Crystallography and Almost Periodicity, Cambridge, 2017, see page 105.
%H A045771 J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (<a href="http://neilsloane.com/doc/sim.txt">Abstract</a>, <a href="http://neilsloane.com/doc/sim.pdf">pdf</a>, <a href="http://neilsloane.com/doc/sim.ps">ps</a>).
%H A045771 <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>.
%H A045771 <a href="/index/Su#sublatts">Index entries for sequences related to sublattices</a>.
%F A045771 Multiplicative with a(2^p) = 1, a(p^e) = (e+1)*p^e + (2*(1+(e*p-e-1)*p^e))/((p-1)^2), p>2. - _Christian G. Bower_, May 21 2005
%F A045771 From _Amiram Eldar_, May 26 2025: (Start)
%F A045771 Dirichlet g.f.: (zeta(s-1)^2 * zeta(s)^2 / zeta(2*s)) * (1 - 1/2^(s-1))^2/(1 + 1/2^s).
%F A045771 Sum_{k=1..n} a(k) ~ (n^2/4)*(log(n) + 2*gamma - 1/2 + 11*log(2)/5 + 2*zeta'(2)/zeta(2) - 2*zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)
%t A045771 Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[1 <= p <= 2, 1, (e + 1) p^e + (2 (1 + (e p - e - 1)*p^e))/((p - 1)^2)]] &, 64] (* _Michael De Vlieger_, Mar 02 2018 *)
%o A045771 (PARI) fp(p, e) = if (p % 2, (e+1)*p^e + 2*(1-(e+1)*p^e+e*p^(e+1))/(p-1)^2, 1);
%o A045771 a(n) = { my(f = factor(n)); prod(i=1, #f~, fp(f[i, 1], f[i, 2]));} \\ _Michel Marcus_, Mar 03 2014
%Y A045771 Cf. A001620, A013662, A035292, A261506, A306016.
%K A045771 nonn,mult
%O A045771 1,3
%A A045771 Michael Baake (baake(AT)miles.math.ualberta.ca)
%E A045771 More terms from _Michel Marcus_, Mar 03 2014