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%I #11 Jul 10 2018 18:52:47
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,24,1,57,170,42,0,618,48,84,0,0,0,84
%N 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).
%C 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)).
%H Andrew Howroyd, <a href="/A316566/b316566.txt">Table of n, a(n) for n = 1..8660</a> (first 40 rows)
%F T(p*q,k) = Sum_{i>0, j>0, k=lcm(i, j)} T(p, i)*T(q, j) for gcd(p, q)=1.
%F T(n,k) = Sum_{d|k} mu(d/k) * A316584(n,k).
%e Triangle begins:
%e 1
%e 1, 3, 2
%e 1, 13, 8, 6, 0, 8, 0, 12
%e 1, 27, 8, 36, 0, 24
%e 1, 31, 20, 152, 24, 20, 0, 40, 0, 24, 0, 40, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 80
%e 1, 55, 26, 24, 0, 98, 0, 48, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24
%e ...
%o (PARI)
%o MatOrder(M)={my(id=matid(#M), k=1, N=M); while(N<>id, k++;N=N*M); k}
%o 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)}
%o for(n=1, 6, print(row(n)))
%Y Row sums are A000252.
%Y Column 2 is A066947.
%Y Cf. A316560, A316564, A316565, A316584.
%K nonn,tabf
%O 1,3
%A _Andrew Howroyd_, Jul 06 2018
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