A227867 Number of Lipschitz quaternions X such that X^2 == 1 (mod n).

1, 8, 14, 32, 32, 112, 58, 32, 110, 256, 134, 448, 184, 464, 448, 32, 308, 880, 382, 1024, 812, 1072, 554, 448, 752, 1472, 974, 1856, 872, 3584, 994, 32, 1876, 2464, 1856, 3520, 1408, 3056, 2576, 1024, 1724, 6496, 1894, 4288, 3520

Offset

1,2

Comments

A quaternion q = a + bi + cj + dk is congruent to 1 (mod n) iff a == 1 (mod n) and b == c == d == 0 (mod n).

Links

Table of n, a(n) for n=1..45.

Wikipedia, Hurwitz quaternion

Mathematica

cuaternios[n_] := Flatten[Table[{{a, -b, d, -c}, {b, a, -c, -d}, {-d, c, a, -b}, {c, d, b, a}}, {a, n}, {b, n}, {c, n}, {d, n}], 3]; invo[n_] := invo[n] = Length@Select[cuaternios[n], Mod[#.# - IdentityMatrix[4], n] == 0*# &]; Table[invo[n], {n, 1, 25}]

Crossrefs

Cf. A227477, A227499, A227628.

Sequence in context: A153340 A144842 A273843 * A254034 A225606 A117132

Adjacent sequences: A227864 A227865 A227866 * A227868 A227869 A227870

Keyword

nonn,mult,more

Author

José María Grau Ribas, Nov 02 2013

Status

approved