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, 4432, 2258, 448, 2746, 6016, 4312, 5888

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..52.

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

Author

José María Grau Ribas, Nov 02 2013

Extensions

More terms from Amiram Eldar, May 06 2024

Status

approved