Symmetriads

Symmetriads

In the range until 1,000,000,000

Evensymmetriades have 2, 4, 6, 8, 10,

Oddsymmetriades have 3, 5, and 7 variations

G_c – the gap in the center of the symmetrical group

* Numbers of groups and pairs do not include ones of special kind of it.

6666666666666666666

In the range until 1,000,000

Evensymmetiads

S2. + V – V. G_c, S4. +V1+V-V-V1.G_c, S6. +V2+V1+V-V-V1-V2.G_c, S8. +V3+V2+V1+V-V-V1-V2-V3.G_c

S10. +V4+V3+V2+V1+V-V-V1-V2-V3-V4.G_c

Designation

Samples

S2. + V – V. G_c

G₁ = g₂     G_c = G₂
Total number in the ranges:
until 100,000   994 in 23 pairs
until 1,000,000   6728 in 36 pairs

106	577	6	4
107	587	10	-4
108	593	6	0

S4. +V1+V-V-V1.G_c

G₁ = g₄ G_{2
=} G₄

G_{c
=} G₃

Total number in the ranges:
until 100,000 129 in 56 pairs
until 1,000,000 709 in 158 pairs

Regular

2451	21851	10	-2
2452	21859	8	-4
2453	21863	4	4
2454	21871	8	2
2455	21881	10	2

Intermediate

37	157	6	0
38	163	6	-2
39	167	4	2
40	173	6	0
41	179	6	-4

Irregular

9	23	4	2
10	29	6	-4
11	31	2	4
12	37	6	-2
13	41	4	-2

S6. +V2+V1+V-V-V1-V2.G_c

G₁ = g₆ G_{2
=} G₆ G_{3
=} G₅

G_{c
=} G₄

Total number in the ranges:
until 100,000 20 in 19 pairs
until 1,000,000 96 in 82 pairs

124	683	6	2
125	691	8	2
126	701	10	-2
127	709	8	2
128	719	10	-2
129	727	8	-2
130	733	6	0

S8. +V3+V2+V1+V-V-V1-V2-V3.G_c

G₁ = g₈ G_{2
=} G₈ G_{3
=} G₇ G_{4
=} G₆

G_{c
=} G₅

Total number in the ranges:
until 100,000 6 in 6 pairs

7	17	4	-2
8	19	2	2
9	23	4	2
10	29	6	-4
11	31	2	4
12	37	6	-2
13	41	4	-2
14	43	2	2
15	47	4	2

S10. +V4+V3+V2+V1+V-V-V1-V2-V3-V4.G_c

G₁ = g₁₀ G_{2
=} G₁₀ G_{3
=} G₉ G_{4
=} G₈ G_{5
=} G₇
G_{c
=} G₆

Total number in the ranges:
until 100,000 2 in 2 pairs
until 1,000,000 5 in 5 pairs

5660	55793	6	0
5661	55799	6	2
5662	55807	8	-2
5663	55813	6	-2
5664	55817	4	-2
5665	55819	2	2
5666	55823	4	2
5667	55829	6	2
5668	55837	8	-2
5669	55843	6	0
5670	55849	6	16

2-ads S2

V⁰ ∩ G_c⁰			V^{~ 0} ∩ G_c^{~ 0}
		666666
V > 0:			V < 0:
V ⁰	G_{c min} = V + 6	G_{1 min} = 6	V ⁰	G_{c min} = 6	G_{1 min} = 12
V⁺	G_{c min} = V + 4	G_{1 min} = 4	V⁺	G_{c min} = 2	G_{1 min} = 4
V^-	G_{c min} = V + 2	G_{1 min} = 2	V^-	G_{c min} = 2	G_{1 min} = 6
666666 For every list of 2-ades having the same value of V and sorted by G_c
{ V ⁰				∩ (P_{n + 1} – P_n)² }
{ V ^~0 ∩ ( G_{c( n + 1)} = G_{c n} U G_{c( n+ 1)} = G_{c n} + 6 )				∩ (P_{n + 1} – P_n)⁰}
{ V ^~0 ∩ ~(G_{c (n + 1)} = G_{c n} U G_{c( n+ 1)} = G_{c n} + 6 )				∩ (P_{n + 1} – P_n)^~0}

666666

Distribution of the number of S2 in pairs
in the range until 100,000

All pairs form 3 series:

The pairs having V⁺ form the major series Y ⁺= 347e^-^{0.124 V}

The pairs having V^- form the medium series Y ^- = 156e^{-
0.105
V}

The pairs having V⁰ form the minor series Y ⁰ = 122e^-

0.131V

Oddsymmetriads

S3. + V1 +0 – V1. G_c, S5. + V1 +V2+0 –V2 –V1. G_c
G_n Î G⁰and V_n Î V⁰

Designation

Samples

S3. + V1 +0 – V1. G_c
G₁ = g₃ G₂ = G₃
G_{c
=} G₂
Total number in the ranges:
until 100,000 8 in 2 pairs
until 1,000,000 54 in 4 pairs

2137	18719	6	6
2138	18731	12	0
2139	18743	12	-6
2140	18749	6	2

S5. +V1+ V2 +0 – V2- V1. G_c
G₁ = g₅ G₂ = G₅ G₃ = G₄
G_{c

=} G₃
Total number in the ranges:
until 100,000 0
until 1,000,000 1

For S3

V_c = 0 Î V⁰

® G₂ Î G ⁰ and G₃ = g₂ Î G ⁰

® V1 Î V ^~k and -V1 Î V ^~k

But -V1 º R( V1)

® V1 Î V ⁰ and -V1 Î V ⁰

and

G₁ Î G ⁰ and g₃ Î G ⁰

There is the same for S5 and any oddsymmetriads

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