A239395 Twice prime nonnegative Hurwitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

2, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 3, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 0, 2, 2, 2, 4, 2, 0, 0, 4, 0, 2, 0, 4, 0, 0, 2, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 2, 4, 0, 0, 2, 0, 4, 0

Offset: 1

Author

T. D. Noe, Mar 21 2014

Keywords

• nonn
• nice
• tabf

Comments

The vectors are multiplied by 2 because a Hurwitz quaternion can have half-integer integer components. The norms of quaternions are (rational) primes 2, 3, 5, 7, 11, ... A quaternion is commonly written a + b*i + c*j + d*k, where 1, i, j, and k are units.

Links

• T. D. Noe, Table of n, a(n) for n = 1..2612 (4-vectors)
• Wikipedia, Hurwitz quaternion.

Crossrefs

Cf. A239393 (Lipschitz quaternions).

Cf. A239396 (number of Hurwitz quaternions having norm prime(n)).

Programs

Mathematica

(* first << Quaternions` *) mx = 5; lst = Flatten[Table[{a, b, c, d}/2, {a, 0, mx}, {b, 0, mx}, {c, 0, mx}, {d, 0, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; 2*Sort[q, Norm[#1] < Norm[#2] &]