A227477 -id:A227477 - cp's OEIS frontend

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

José María Grau Ribas, Nov 02 2013

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).

Wikipedia, Hurwitz quaternion.

Cf. A227477, A227499, A227628.

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}]

More terms from Amiram Eldar, May 06 2024

A227628 Number of Lipschitz quaternions X such that X^2 == X (mod n).

Original entry on oeis.org

1, 2, 14, 2, 32, 28, 58, 2, 110, 64, 134, 28, 184, 116, 448, 2, 308, 220, 382, 64, 812, 268, 554, 28, 752, 368, 974, 116, 872, 896, 994, 2, 1876, 616, 1856, 220, 1408, 764, 2576, 64, 1724, 1624, 1894, 268, 3520, 1108, 2258, 28, 2746, 1504

Offset: 1

José María Grau Ribas, Jul 18 2013

C. J. Miguel and R. Serodio, On the Structure of Quaternion Rings over Zp, International Journal of Algebra, Vol. 5, 2011, no. 27, pp. 1313-1325.

Cf. A000118, A227477, A227499.

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]; cuater[n_] := Length@Select[cuaternios[n], Mod[#.# - #, n] == 0*# &]; Table[cuater[n],{n,1,100}]

A229292 Exponent of the group of 2 X 2 invertible matrices over Z/nZ.

Original entry on oeis.org

1, 6, 24, 30, 120, 24, 336, 126, 240, 120, 1320, 120, 2184, 336, 120, 510, 4896, 240, 6840, 120, 336, 1320, 12144, 504, 3120, 2184, 2184, 1680, 24360, 120, 29760, 2046, 1320, 4896, 1680, 240, 50616, 6840, 2184, 2520, 68880, 336, 79464, 1320, 240, 12144

Offset: 1

José María Grau Ribas, Nov 02 2013

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000252, A227477, A316566.

ex[p_, s_] := LCM[p(p^(2 s) - 1), p - 1]; ex[1] := 1; ex[n_] := {aux = 1; Do[aux = LCM[aux, ex[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length[fa[n]]}]; aux}[[1]];Table[ex[n], {n, 1, 111}]

a(n)=if(n==1,return(1)); my(f=factor(n)); lcm(vector(#f~,i, f[i,1]*lcm((f[i,1]^(2*f[i,2])-1), f[i,1]-1))) \\ Charles R Greathouse IV, Nov 13 2013

a(p^s) = lcm(p*(p^(2*s) - 1), p - 1); if gcd(m,n)=1 then a(n*m) = lcm(a(n), a(m)).

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

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A227628 Number of Lipschitz quaternions X such that X^2 == X (mod n).

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A229292 Exponent of the group of 2 X 2 invertible matrices over Z/nZ.

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