A239393 Nonnegative prime Lipschitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 1, 2, 0, 0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 1, 2

Offset: 1

Author

T. D. Noe, Mar 21 2014

Keywords

• nonn
• nice

Comments

A Lipschitz quaternion has all 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.

Examples

			The first six nonnegative prime Lipschitz quaternions are 1+i, 1+j, 1+k, i+j, i+k, and j+k.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..2586 (4-vectors)
• Wikipedia , Hurwitz quaternion

Crossrefs

Cf. A239394 (number of Lipschitz quaternions having norm prime(n)).

Cf. A239395 (Hurwitz quaternions).

Programs

Mathematica

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