In the range until 1,000,000,000

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

Oddsymmetriades have                 3, 5, and  7 variations

Gc 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

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

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

Designation

Samples

S2. + V V. Gc

G1 = g2     Gc = 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.Gc

G1 = g4       G2 = G4

Gc = 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.Gc

G1 = g6     G2 = G6     G3 = G5

Gc = 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.Gc

G1 = g8     G2 = G8     G3 = G7     G4 = G6

Gc = 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.Gc

G1 = g10     G2 = G10     G3 = G9     G4 = G8     G5 = 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

 V 0 ∩ Gc0 V ~ 0 ∩ Gc~ 0 666666 V > 0: V < 0: V 0 Gc min = V + 6 G1 min = 6 V 0 Gc min = 6 G1 min = 12 V+ Gc min = V + 4 G1 min = 4 V+ Gc min = 2 G1 min = 4 V- Gc min = V + 2 G1 min = 2 V- Gc min = 2 G1 min = 6 666666   For every list of 2-ades having the same value of V and sorted by Gc { V 0 ∩ (Pn + 1 – Pn)2 } { V ~0 ∩ ( Gc( n + 1) = Gc n U Gc( n+ 1) = Gc n + 6 ) ∩ (Pn + 1 – Pn)0} { V ~0 ∩ ~(Gc (n + 1) = Gc n U Gc( n+ 1) = Gc n + 6 ) ∩ (Pn + 1 – Pn)~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

S3. + V1 +0 V1. Gc, S5. + V1 +V2+0 V2 V1. Gc
Gn Î G 0         and        Vn Î V 0

Designation

Samples

S3. + V1 +0 V1. Gc
G
1 = 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. Gc
G
1 = g5  G2 = G5   G3 = G4
G
c = G3
Total number in the ranges:
until 100,000      0
until 1,000,000   1

For  S3

Vc = 0
Î V 0

®        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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