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A327570 a(n) = n*phi(n)^2, phi = A000010. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..48.

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

EXAMPLE

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.

PROG

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

CROSSREFS

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

Sequence in context: A295821 A113802 A274890 * A057123 A134833 A057827

Adjacent sequences:  A327566 A327568 A327569 * A327571 A327572 A327573

KEYWORD

nonn,easy,mult

AUTHOR

Jianing Song, Sep 17 2019

STATUS

approved

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Last modified April 6 11:38 EDT 2020. Contains 333273 sequences. (Running on oeis4.)