A264081 - OEIS

%I #23 Jan 15 2017 13:12:40

%S 0,0,1,1,1,2,1,1,1,8,1,2,1,8,14,1,1,8,1,16,7,8,1,14,1,8,1,26,1,22,1,1,

%T 14,8,18,26,1,8,1,8,1,14,1,26,5,8,1

%N The sum of the 2 X 2 idempotent matrices over Z/nZ is congruent to {{a(n),0}, {0,a(n)}} (mod n).

%e The 2 X 2 idempotent matrices over Z/3Z are {{0, 0}, {0, 0}}, {{0, 0}, {0,1}}, {{0, 0}, {1, 1}}, {{0, 0}, {2, 1}}, {{0, 1}, {0, 1}}, {{0, 2}, {0, 1}}, {{1, 0}, {0, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 0}}, {{1, 0}, {2, 0}}, {{1, 1}, {0, 0}}, {{1, 2}, {0, 0}}, {{2, 1}, {1, 2}}, {{2, 2}, {2, 2}}. Their sum is {{10, 9}, {9, 10}} == 1*{{1, 0}, {0, 1}} (mod 3) and therefore a(3) = 1.

%t K[n_] := K[n] = Mod[Sum[If[ Mod[{{a, b}, {c, d}}.{{a, b}, {c, d}} - {{a, b}, {c, d}}, n] == 0{{a, b}, {c, d}}, {{a, b}, {c, d}}, 0], {a, n}, {b, n}, {c, n}, {d, n}], n]; Table[K[n][[1, 1]], {n, 1, 22}]

%o (PARI) a(n) = lift(sum(i=0, n-1, sum(j=0, n-1, sum(k=0, n-1, sum(l=0, n-1, m = Mod([i,j;k,l], n); if ((m^2 == m), m[1,1])))))); \\ _Michel Marcus_, Apr 04 2016

%Y Cf. A226756.

%K nonn,more

%O 1,6

%A _José María Grau Ribas_, Nov 03 2015