In order to answer this question, I came up with a way to generate the reciprocals of aurifeuillan factors of $n$-th cyclotomic polynomials for odd prime $n$. If $n=1\pmod 4$, then $\Phi_n(nx^2)$ has an aurifeuillan factorization, otherwise, $\Phi_{2n}(nx^2)$ has one. If $K$ is the field of $n$-th roots of unity, and $r$ is an $n$-th root of unity, then the minimal polynomials of the expressions below are one of the aurifeuillan factors of $n$-th or $2n$-th cyclotomic polynomial:
PARI/GP:
(09:17) gp > minpoly(Mod(x + 1,polcyclo(6)))
%34 = x^2 - 3*x + 3
(09:17) gp > minpoly(Mod(x^3 - x^2 - x + 1,polcyclo(5)))
%35 = x^4 - 5*x^3 + 15*x^2 - 25*x + 25
(09:17) gp > minpoly(Mod(x^5 - x^4 - x^3 - x^2 - x + 1,polcyclo(14)))
%36 = x^6 - 7*x^5 + 21*x^4 - 49*x^3 + 147*x^2 - 343*x + 343
(09:17) gp > minpoly(Mod(x^9 + x^8 + x^7 - x^6 + x^5 + x^4 - x^3 + x^2 + x + 1,polcyclo(22)))
%37 = x^10 - 11*x^9 + 55*x^8 - 121*x^7 - 121*x^6 + 1331*x^5 - 1331*x^4 - 14641*x^3 + 73205*x^2 - 161051*x + 161051
(09:18) gp > minpoly(Mod(x^11 - x^10 + x^9 + x^8 - x^7 - x^6 - x^5 - x^4 + x^3 + x^2 - x + 1,polcyclo(13)))
%38 = x^12 - 13*x^11 + 91*x^10 - 507*x^9 + 2535*x^8 - 10985*x^7 + 41743*x^6 - 142805*x^5 + 428415*x^4 - 1113879*x^3 + 2599051*x^2 - 4826809*x + 4826809
(09:18) gp > minpoly(Mod(x^15 + x^14 - x^13 + x^12 - x^11 - x^10 - x^9 + x^8 + x^7 - x^6 - x^5 - x^4 + x^3 - x^2 + x + 1,polcyclo(17)))
%39 = x^16 - 17*x^15 + 153*x^14 - 867*x^13 + 3179*x^12 - 4913*x^11 - 24565*x^10 + 250563*x^9 - 1252815*x^8 + 4259571*x^7 - 7099285*x^6 - 24137569*x^5 + 265513259*x^4 - 1231016019*x^3 + 3693048057*x^2 - 6975757441*x + 6975757441
(09:18) gp > minpoly(Mod(x^17 + x^16 - x^15 - x^14 + x^13 - x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 - x^5 + x^4 - x^3 - x^2 + x + 1,polcyclo(38)))
%40 = x^18 - 19*x^17 + 171*x^16 - 1083*x^15 + 6137*x^14 - 34295*x^13 + 185193*x^12 - 912247*x^11 + 4039951*x^10 - 17332693*x^9 + 76759069*x^8 - 329321167*x^7 + 1270238787*x^6 - 4469358695*x^5 + 15195819563*x^4 - 50950689123*x^3 + 152852067369*x^2 - 322687697779*x + 322687697779
(09:18) gp > minpoly(Mod(x^21 - x^20 + x^19 - x^18 - x^17 - x^16 - x^15 - x^14 + x^13 + x^12 - x^11 - x^10 + x^9 + x^8 - x^7 - x^6 - x^5 - x^4 - x^3 + x^2 - x + 1,polcyclo(46)))
%41 = x^22 - 23*x^21 + 253*x^20 - 1587*x^19 + 4761*x^18 + 12167*x^17 - 231173*x^16 + 1399205*x^15 - 4197615*x^14 - 6436343*x^13 + 160908575*x^12 - 1036251223*x^11 + 3700897225*x^10 - 3404825447*x^9 - 51072381705*x^8 + 391554926405*x^7 - 1487908720339*x^6 + 1801152661463*x^5 + 16210373953167*x^4 - 124279533640947*x^3 + 455691623350139*x^2 - 952809757913927*x + 952809757913927
The inverses of the minimal polynomials formatted in the code are divisors of $\Phi_n(nx^2)$ or $\Phi_{2n}(nx^2)$.
Does anyone know a relation between the generators of the minimal polynomials:
$r+1$
$r^3 - r^2 - r + 1$
$r^5 - r^4 - r^3 - r^2 - r + 1$
$r^9 + r^8 + r^7 - r^6 + r^5 + r^4 - r^3 + r^2 + r + 1$
$r^{11} - r^{10} + r^9 + r^8 - r^7 - r^6 - r^5 - r^4 + r^3 + r^2 - r + 1$
$......$