A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Jun 09 2000

Keywords

• nonn
• easy
• nice

Comments

Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units.

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

References

• L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

Links

• R. J. Mathar, Table of n, a(n) for n = 0..10000
• L. E. Dickson, Algebras and Their Arithmetics, U. Chicago Press, 1923, Section 91.

Crossrefs

Cf. A055669 A055670 (zeros removed), A055671, A385603 (equivalence up to left-and-right multiplication).

Programs

Mathematica

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 *)

Formula

a(n) = A055671(n)/24.

a(n) = A000593(n) * A010051(n). - R. J. Mathar, Aug 02 2025

Extensions

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.