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A316566 Triangle read by rows: T(n,k) is the number of elements of the group GL(2, Z(n)) with order k, 1 <= k <= A316565(n). 9
1, 1, 3, 2, 1, 13, 8, 6, 0, 8, 0, 12, 1, 27, 8, 36, 0, 24, 1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80, 1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 57, 170, 42, 0, 618, 48, 84, 0, 0, 0, 84 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For coprime p,q the group GL(p*q, Z(n)) is isomorphic to the direct product of the two groups GL(p, Z(n)) and GL(q, Z(n)).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..8660 (first 40 rows)

FORMULA

T(p*q,k) = Sum_{i>0, j>0, k=lcm(i, j)} T(p, i)*T(q, j) for gcd(p, q)=1.

T(n,k) = Sum_{d|k} mu(d/k) * A316584(n,k).

EXAMPLE

Triangle begins:

  1

  1, 3, 2

  1, 13, 8, 6, 0, 8, 0, 12

  1, 27, 8, 36, 0, 24

  1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80

  1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24

  ...

PROG

(PARI)

MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++; N=N*M); k}

row(n)={my(L=List()); for(a=0, n-1, for(b=0, n-1, for(c=0, n-1, for(d=0, n-1, my(M=Mod([a, b; c, d], n)); if(gcd(lift(matdet(M)), n)==1, my(t=MatOrder(M)); while(#L<t, listput(L, 0)); L[t]++ ))))); Vec(L)}

for(n=1, 6, print(row(n)))

CROSSREFS

Row sums are A000252.

Column 2 is A066947.

Cf. A316560, A316564, A316565, A316584.

Sequence in context: A059438 A156628 A104980 * A134090 A132845 A129652

Adjacent sequences:  A316563 A316564 A316565 * A316567 A316568 A316569

KEYWORD

nonn,tabf

AUTHOR

Andrew Howroyd, Jul 06 2018

STATUS

approved

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Last modified September 15 12:22 EDT 2019. Contains 327078 sequences. (Running on oeis4.)