User:Lériendil/Square and triangle superparticulars by prime subgroup

Some shorthand notation used here:

• Sk stands for k^2/[(k-1)(k+1)] by standard convention.
• Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2].
• Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
• Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.

This list eventually aims to be complete to the semi-41-limit, i.e. the class of subgroups with at most one prime greater than 41.

2- and 3-prime subgroups (2.3 and 2.3.p)

Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.

2-prime subgroup (2.3)

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S2	2.3	4/3	[2 -1⟩
S3	2.3	9/8	[-3 2⟩

Triangle-particular	Subgroup	Comma
		Ratio	Smonzo
T2	2.3	3/2	[-1 1⟩

3-prime subgroups (2.3.p)

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S4	2.3.5	16/15	[4 -1 -1⟩
S5	2.3.5	25/24	[-3 -1 2⟩
S9	2.3.5	81/80	[-4 4 -1⟩
S7	2.3.7	49/48	[-4 -1 2⟩
S8	2.3.7	64/63	[6 -2 -1⟩
S17	2.3.17	289/288	[-5 -2 2⟩

Triangle-particular	Subgroup	Comma
		Ratio	Smonzo
T3	2.3.5	6/5	[1 1 -1⟩
T4	2.3.5	10/9	[1 -2 1⟩
T7	2.3.7	28/27	[2 -3 1⟩

4-prime subgroups

Semi-5-limit (L5.p)

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S6	2.3.5.7	36/35	[2 2 -1 -1⟩
S15	2.3.5.7	225/224	[-5 2 2 -1⟩
S49	2.3.5.7	2401/2400	[-5 -1 -2 4⟩
S10	2.3.5.11	100/99	[2 -2 2 -1⟩
S11	2.3.5.11	121/120	[-3 -1 -1 2⟩
S25	2.3.5.13	625/624	[-4 -1 4 -1⟩
S26	2.3.5.13	676/675	[2 -3 -2 2⟩
S16	2.3.5.17	256/255	[8 -1 -1 -1⟩
S19	2.3.5.19	361/360	[-3 -2 -1 2⟩
S24	2.3.5.23	576/575	[6 2 -2 -1⟩
S81	2.3.5.41	6561/6560	[-5 8 -1 -1⟩
S80	2.3.5.79	6400/6399	[8 -4 2 -1⟩

Triangle-particular	Subgroup	Comma
		Ratio	Smonzo
T5	2.3.5.7	15/14	[-1 1 1 -1⟩
T9	2.3.5.11	45/44	[-2 2 1 -1⟩
T10	2.3.5.11	55/54	[-1 -3 1 1⟩
T25	2.3.5.13	325/324	[-2 -4 2 1⟩