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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.
Sequence in context: A059438 A156628 A104980 * A134090 A132845 A129652
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Jul 06 2018
STATUS
approved

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Last modified July 13 11:43 EDT 2024. Contains 374282 sequences. (Running on oeis4.)