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Number of prime nonnegative Lipschitz quaternions having norm prime(n).
3

%I #10 Mar 31 2014 17:40:20

%S 6,4,12,4,12,16,24,16,12,36,16,28,48,28,24,48,48,52,40,36,52,40,60,84,

%T 64,96,52,72,76,84,64,96,96,88,120,76,100,88,84,132,120,124,96,112,

%U 132,100,124,112,144,148,156,120,160,168,180,132,204,136,160,204

%N Number of prime nonnegative Lipschitz quaternions having norm prime(n).

%C For n > 1, there are prime(n) + 1 more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions. - T. D. Noe, Mar 31 2014

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

%e The six prime nonnegative Lipschitz quaternions having norm 2 are 1+i, 1+j, 1+k, i+j, i+k, and j+k.

%t (* first << Quaternions` *) mx = 17; 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] &]; q2 = Sort[q, Norm[#1] < Norm[#2] &]; Transpose[Tally[(Norm /@ q2)^2]][[2]]

%Y Cf. A239393 (prime Lipschitz quaternions).

%Y Cf. A239395 (prime Hurwitz quaternions).

%K nonn

%O 1,1

%A _T. D. Noe_, Mar 21 2014