A240068 Number of prime Lipschitz quaternions having norm prime(n).

24, 32, 48, 64, 96, 112, 144, 160, 192, 240, 256, 304, 336, 352, 384, 432, 480, 496, 544, 576, 592, 640, 672, 720, 784, 816, 832, 864, 880, 912, 1024, 1056, 1104, 1120, 1200, 1216, 1264, 1312, 1344, 1392, 1440, 1456, 1536, 1552, 1584, 1600, 1696, 1792

Offset: 1

Author

T. D. Noe, Apr 01 2014

Keywords

• nonn

Comments

This sequence counts all prime Lipschitz quaternions having a given norm; A239394 counts only the prime nonnegative Lipschitz quaternions.

Links

• Wikipedia, Hurwitz quaternion

Crossrefs

Cf. A239393 (prime Lipschitz quaternions), A239394.

Cf. A055669 (number of prime Hurwitz quaternions of norm prime(n)).

Programs

Mathematica

(* first << Quaternions` *)
mx = 17; lst = Flatten[Table[{a, b, c, d}, {a, -mx, mx}, {b, -mx, mx}, {c, -mx, mx}, {d, -mx, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; q2 = Sort[q, Norm[#1] < Norm[#2] &]; Take[Transpose[Tally[(Norm /@ q2)^2]][[2]], mx]

Formula

a(n) = 8 * (prime(n) + 1) = 8 * A008864(n).