A327570 a(n) = n*phi(n)^2, phi = A000010.

1, 2, 12, 16, 80, 24, 252, 128, 324, 160, 1100, 192, 1872, 504, 960, 1024, 4352, 648, 6156, 1280, 3024, 2200, 11132, 1536, 10000, 3744, 8748, 4032, 22736, 1920, 27900, 8192, 13200, 8704, 20160, 5184, 47952, 12312, 22464, 10240, 65600, 6048, 75852, 17600, 25920, 22264, 99452, 12288

Offset: 1

Author

Jianing Song, Sep 17 2019

Keywords

• nonn
• easy
• mult

Comments

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}}.

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.)

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000252, A002618, A174824, A011379, A065464.

Programs

Mathematica

Table[n * EulerPhi[n]^2, {n, 1, 100}] (* Amiram Eldar, Sep 19 2020 *)

PARI

a(n) = n*eulerphi(n)^2

Formula

Multiplicative with a(p^e) = (p-1)^2*p^(3e-2).

a(n) = A000010(n)*A002618(n).

a(p) = A011379(p-1) for p prime. - Peter Luschny, Sep 17 2019

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

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