A264081 The sum of the 2 X 2 idempotent matrices over Z/nZ is congruent to {{a(n),0}, {0,a(n)}} (mod n).

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, 14, 8, 18, 26, 1, 8, 1, 8, 1, 14, 1, 26, 5, 8, 1

Offset

1,6

Links

Table of n, a(n) for n=1..47.

Example

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.

Mathematica

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}]

Prog

(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

Crossrefs

Cf. A226756.

Sequence in context: A171660 A157117 A322143 * A061538 A123602 A208896

Adjacent sequences: A264078 A264079 A264080 * A264082 A264083 A264084

Keyword

nonn,more

Author

José María Grau Ribas, Nov 03 2015

Status

approved