A239395 - OEIS

%I #15 Sep 19 2023 06:26:13

%S 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,

%T 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,

%U 3,3,1,1,3,1,3,1,3,1,1,3,2,4,0,0,2,0,4,0

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

%C 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.

%H T. D. Noe, <a href="/A239395/b239395.txt">Table of n, a(n) for n = 1..2612</a> (4-vectors)

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hurwitz_quaternion">Hurwitz quaternion</a>.

%t (* 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] &]

%Y Cf. A239393 (Lipschitz quaternions).

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

%K nonn,nice,tabf

%O 1,1

%A _T. D. Noe_, Mar 21 2014