Ben Whitmore, <a href="/A339122/b339122_1.txt">Table of n, a(n) for n = 1..73</a>

proposed

approved

editing

proposed

Module[{z}, z = pN[p1] pN[p2]; {{LCM[ord[p1], ord[p2]],

{{LCM[ord[p1], ord[p2]], z oriN[p1, o1][[1]] oriN[p2, o2][[1]]}, {{LCM[ord[p1] o1,

{{LCM[ord[p1] o1, ord[p2]], z oriN[p1, o1][[2]] oriN[p2, o2][[1]]}}, {{LCM[ord[p1], ord[p2] o2], z oriN[p1, o1][[1]] oriN[p2, o2][[2]], }},

{{LCM[ord[p1] o1, ord[p2] o2], z oriN[p1, o1][[2]] oriN[p2, o2][[12]] }}, {{LCM[ord[p1}],

ord[p2] o2],

z oriN[p1, o1][[1]] oriN[p2, o2][[2]] }}, {{LCM[ord[p1] o1,

ord[p2] o2], z oriN[p1, o1][[2]] oriN[p2, o2][[2]] }}}]

For[j = 1, j <= Length[ee], j++,

For[j = 1, j <= Length[eo], j++,

pN[p_] := Total[p]!/Times@@p/Times@@Factorial[Flatten[Tally[p]][[2 ;; ;; 2]]]

oddQ[p_] := OddQ[Total[p - 1]]

ord[p_] := LCM @@ p

oriN[p_, o_] := Module[{i, t, a = 0, ns = 0, s = 0, r}, t = ord[p]/p;

For[i = 1, i <= Length[p], i++,

If[Mod[t[[i]], o] == 0, a += p[[i]], ns += 1; s += p[[i]]]];

{If[a == 0, r = o^(s - ns), r = o^a o^(s - ns - 1)], o^(a + s - 1) - r}]

val[p1_, o1_, p2_, o2_] :=

Module[{z}, z = pN[p1] pN[p2]; {{LCM[ord[p1], ord[p2]],

z oriN[p1, o1][[1]] oriN[p2, o2][[1]]}, {{LCM[ord[p1] o1,

ord[p2]],

z oriN[p1, o1][[2]] oriN[p2, o2][[1]] }}, {{LCM[ord[p1],

ord[p2] o2],

z oriN[p1, o1][[1]] oriN[p2, o2][[2]] }}, {{LCM[ord[p1] o1,

ord[p2] o2], z oriN[p1, o1][[2]] oriN[p2, o2][[2]] }}}]

p8 = IntegerPartitions[8]; p12 = IntegerPartitions[12];

ce = Select[p8, ! oddQ[#] &]; co = Select[p8, oddQ[#] &];

ee = Select[p12, ! oddQ[#] &]; eo = Select[p12, oddQ[#] &];

res = {}; max = 0;

For[i = 1, i <= Length[ce], i++,

For[j = 1, j <= Length[ee], j++,

AppendTo[res, val[ce[[i]], 3, ee[[j]], 2]]]]

For[i = 1, i <= Length[co], i++,

For[j = 1, j <= Length[eo], j++,

AppendTo[res, val[co[[i]], 3, eo[[j]], 2]]]]

p = Partition[res // Flatten, 2]; c // Clear;

For[i = 1, i <= Length[p], i++,

If [IntegerQ[c[p[[i, 1]]]], c[p[[i, 1]]] += p[[i, 2]],

c[p[[i, 1]]] = p[[i, 2]]]; If[p[[i, 1]] > max, max = p[[i, 1]]]];

Select[Table[c[i], {i, 1, max}], IntegerQ[#] &] (* Herbert Kociemba, Jun 30 2022 *)

proposed

editing

editing

proposed

Corrected a(10) corrected by Ben Whitmore, Jun 27 2022

proposed

editing

editing

proposed

(Python) # See post #11 in SpeedSolving Puzzles Community link.

1, 170911549183, 33894540622394, 4346957030144256, 133528172514624, 140621059298755526, 153245517148800, 294998638981939200, 55333752398428896, 65250897836352192, 34178553690432192, 44590694400, 2330232827455554048, 23298374383021440, 14385471333209856, 150731886270873600

The most common order is 60, with a(33) = 4601524692892925952 4199961633799421952 elements, or about 109.6471% of the group.

Ben Whitmore, <a href="/A339122/b339122_1.txt">Table of n, a(n) for n = 1..73</a>

Jaap's Puzzle Page, Tomas Rokicki, <a href="https://www.jaapschspeedsolving.netcom/puzzlesthreads/cubic3average-scramble-loop-length.htm85241/#p35post-1451996">Order of elementsSpeedSolving Puzzles Community</a>, Cubic Circular, Issue 3 & 4, Spring & Summer 1982, p. 35.

Corrected by Ben Whitmore, Jun 27 2022

approved

editing