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 A327570 #28 Nov 05 2022 08:18:21
%S A327570 1,2,12,16,80,24,252,128,324,160,1100,192,1872,504,960,1024,4352,648,
%T A327570 6156,1280,3024,2200,11132,1536,10000,3744,8748,4032,22736,1920,27900,
%U A327570 8192,13200,8704,20160,5184,47952,12312,22464,10240,65600,6048,75852,17600,25920,22264,99452,12288
%N A327570 a(n) = n*phi(n)^2, phi = A000010.
%C A327570 a(n) is the order of the group consisting of all upper-triangular (or equivalently, lower-triangular) matrices in GL(2, Z_n). That is to say, a(n) = |G_n|, where G_n = {{{a, b}, {0, d}} : gcd(a, n) = gcd(d, n) = 1}. The group G_n is well-defined because the product of two upper-triangular matrices is again an upper-triangular matrix. For example,{{a, b}, {0, d}} * {{x, y}, {0, z}} = {{a*x, a*y+b*z}, {0, d*z}}.
%C A327570 The exponent of G_n (i.e., the least positive integer k such that x^k = e for all x in G_n) is A174824(n). (Note that {{1, 1}, {0, 1}} is an element with order n and there exists some r such that {{r, 0}, {0, r}} is an element with order psi(n), psi = A002322. It is easy to show that x^lcm(n, psi(n)) = Id = {{1, 0}, {0, 1}} for all x in G_n.)
%C A327570 If only upper-triangular matrices in SL(2, Z_n) are wanted, we get a group of order n*phi(n) = A002618(n) and exponent A174824(n).
%H A327570 Amiram Eldar, <a href="/A327570/b327570.txt">Table of n, a(n) for n = 1..10000</a>
%F A327570 Multiplicative with a(p^e) = (p-1)^2*p^(3e-2).
%F A327570 a(n) = A000010(n)*A002618(n).
%F A327570 a(p) = A011379(p-1) for p prime. - _Peter Luschny_, Sep 17 2019
%F A327570 Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/((p-1)^3 * (p^2 + p + 1))) = 1.7394747912949637836019917301710010334604379331855033150372654868327481539... - _Vaclav Kotesovec_, Sep 20 2020
%F A327570 Sum_{k=1..n} a(k) ~ c * n^4, where c = (1/4) * Product_{p prime} (1 - (2*p-1)/p^3) = A065464 / 4 = 0.1070623764... . - _Amiram Eldar_, Nov 05 2022
%e A327570 G_3 = {{{1, 0}, {0, 1}}, {{1, 1}, {0, 1}}, {{1, 2}, {0, 1}}, {{1, 0}, {0, 2}}, {{1, 1}, {0, 2}}, {{1, 2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 1}, {0, 2}}, {{2, 2}, {0, 2}}} with order 12, so a(3) = 12.
%t A327570 Table[n * EulerPhi[n]^2, {n, 1, 100}] (* _Amiram Eldar_, Sep 19 2020 *)
%o A327570 (PARI) a(n) = n*eulerphi(n)^2
%Y A327570 Cf. A000010, A000252, A002618, A174824, A011379, A065464.
%K A327570 nonn,easy,mult
%O A327570 1,2
%A A327570 _Jianing Song_, Sep 17 2019