|
%I #11 Jul 10 2018 18:44:14
%S 1,1,3,2,1,1,8,6,0,8,1,7,8,24,0,8,1,1,20,30,24,20,0,0,0,24,1,7,26,24,
%T 0,74,0,0,0,0,0,12,1,1,56,42,0,56,48,84,0,0,0,0,0,48,1,15,32,144,0,96,
%U 0,96,1,1,98,54,0,98,0,0,144,0,0,108,0,0,0,0,0,144
%N Triangle read by rows: T(n,k) is the number of elements of the group SL(2, Z(n)) with order k, 1 <= k <= A316563(n).
%C For coprime p,q the group SL(p*q, Z(n)) is isomorphic to the direct product of the two groups SL(p, Z(n)) and SL(q, Z(n)).
%H Andrew Howroyd, <a href="/A316564/b316564.txt">Table of n, a(n) for n = 1..3478</a> (first 60 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) A316586(n,d).
%e Triangle begins:
%e 1;
%e 1, 3, 2;
%e 1, 1, 8, 6, 0, 8;
%e 1, 7, 8, 24, 0, 8;
%e 1, 1, 20, 30, 24, 20, 0, 0, 0, 24;
%e 1, 7, 26, 24, 0, 74, 0, 0, 0, 0, 0, 12;
%e 1, 1, 56, 42, 0, 56, 48, 84, 0, 0, 0, 0, 0, 48;
%e 1, 15, 32, 144, 0, 96, 0, 96;
%e 1, 1, 98, 54, 0, 98, 0, 0, 144, 0, 0, 108, 0, 0, 0, 0, 0, 144;
%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(matdet(M)==1, my(t=MatOrder(M)); while(#L<t,listput(L,0)); L[t]++ ))))); Vec(L)}
%o for(n=1, 9, print(row(n)));
%Y Column 2 is A316553.
%Y Row sums are A000056.
%Y Cf. A316537, A316566, A316586.
%K nonn,tabf
%O 1,3
%A _Andrew Howroyd_, Jul 06 2018
|
