This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A055672 #24 Aug 05 2025 07:50:21
%S A055672 0,0,1,4,0,6,0,8,0,0,0,12,0,14,0,0,0,18,0,20,0,0,0,24,0,0,0,0,0,30,0,
%T A055672 32,0,0,0,0,0,38,0,0,0,42,0,44,0,0,0,48,0,0,0,0,0,54,0,0,0,0,0,60,0,
%U A055672 62,0,0,0,0,0,68,0,0,0,72,0,74,0,0,0,0,0,80,0,0,0,84,0,0,0,0,0,90
%N A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.
%C A055672 Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units.
%C A055672 The extension desired below does not exist in the following sense: Let q1 ~R q2 be the equivalence defined by q1 = q2*u (i.e. if q1 and q2 any two HQ's, a unit u exists that solves this). Let q1 ~L q2 be the equivalence defined by q1 = u*q2 (i.e. if q1 and q2 any two HQ's a unit u exists that solves this.) If we define a relation ~RL such that q1 ~RL q2 means (q1 ~R q2 or q1 ~L q2), this relation is not transitive, i.e., not an equivalence. Cause: q1 = q2*u1, q2 = u2*q3, i.e., q1 ~R q2, q2 ~L q3 does not always have a solution with either q1 = q3*u3 or q1= u3*q3. There are pairs of u1 and u2 out of the 24*24 cases where q1 ~L q3 or q1 ~L q3 cannot be solved with any u3. - _R. J. Mathar_, Aug 05 2025
%D A055672 L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.
%H A055672 R. J. Mathar, <a href="/A055672/b055672.txt">Table of n, a(n) for n = 0..10000</a>
%H A055672 L. E. Dickson, <a href="https://ia600105.us.archive.org/28/items/algebrastheirari00dick_0/algebrastheirari00dick_0.pdf">Algebras and Their Arithmetics</a>, U. Chicago Press, 1923, Section 91.
%F A055672 a(n) = A055671(n)/24.
%F A055672 a(n) = A000593(n) * A010051(n). - _R. J. Mathar_, Aug 02 2025
%t A055672 A055671[n_] := If[PrimeQ[n], Reduce[a^2 + b^2 + c^2 + d^2 == 4n, {a, b, c, d}, Integers] // Length, 0]; a[n_] := A055671[n]/24; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Oct 22 2016 *)
%Y A055672 Cf. A055669 A055670 (zeros removed), A055671, A385603 (equivalence up to left-and-right multiplication).
%K A055672 nonn,easy,nice
%O A055672 0,4
%A A055672 _N. J. A. Sloane_, Jun 09 2000
%E A055672 I would also like to get the sequences of inequivalent prime Hurwitz quaternions, where two primes are considered equivalent if they differ by left or right multiplication by one of the 24 units. This will give two more sequences, analogs of A055670 and A055672.