Summarized Results for 100,057 Second Run Zero Omitted

March 1, 2005

General

All: 100,055 Graphs Relatively Prime: 30,239 Graphs Primitive Root: 30,240 Graphs Relatively Prime and Primitive Root: 9,192 Graphs Neither: 48,768 Graphs Not Primitive Root: 69815 Graphs

Largest Cycle: 100,052 (g = 58, 303) Longest Tail: 1,589 (g = 18, 115) Shortest Maximum Cycle: 1 (24 values of g)

Number of Components

Observed All:

760, 744/100, 055 ≈ 7.603

Observed RP:

229, 961/30, 239 ≈ 7.605

Maps

Theoretical:

\frac{1}{2} log n = \frac{1}{2} log 100, 057 ≈ 5.76

Observed Not PR:

396, 228/69, 815 ≈ 5.68

Permutations

Theoretical:

∑_{i=1}^{100,057} \frac{1}{i} ≈ 12.09

Observed PR:

364, 516/30, 240 ≈ 12.05

Number of Cyclic Nodes

Observed All:

3, 041, 638, 223/100, 055 ≈ 30, 399.66

Observed RP:

924, 524, 805/30, 239 ≈ 30, 573.92

Maps

Theoretical:

\sqrt{πn/2} = \sqrt{π100, 057/2} ≈ 396.457

Observed None:

11, 134, 730/48, 768 ≈ 228.32

Observed Not PR:

15, 944, 783/69, 815 ≈ 228.39

Permutations

Theoretical:

100, 056

Observed PR:

3, 025, 693, 440/30, 240 = 100, 056

Observed RPPR:

919, 714, 752/9, 192 = 100, 056

Number of Tail Nodes

Observed All:

6, 969, 564, 912/100, 055 ≈ 69, 657.33

Observed RP:

2, 101, 098, 818/30, 239 ≈ 69, 483.08

Maps

Theoretical:

n − \sqrt{πn/2} = 100, 057 − \sqrt{π100, 057/2} ≈ 99, 660.55

Observed None:

4, 868, 445, 046/48, 768 ≈ 99, 828.68

Observed Not PR:

6, 969, 534, 672/69, 815 ≈ 99, 828.61

Permutations

Theoretical:

1

Observed PR:

30, 240/30, 240 = 1

Observed RPPR:

9, 192/9, 192 = 1

Number of Terminal Nodes

Observed All:

5, 224, 643, 043/100, 055 ≈ 52, 217.71

Observed RP:

1, 572, 707, 711/30, 239 ≈ 52, 009.25

Maps

Theoretical:

e^{−1}n = e^{−1} ∗ 100, 057 ≈ 36, 808.9

Observed None:

3, 651, 914, 284/48, 768 ≈ 74, 883.41

Observed Not PR:

5, 224, 612, 803/69, 815 ≈ 74, 835.10

Permutations

Theoretical:

1

Observed PR:

30, 240/30, 240 = 1

Observed RPPR:

9, 192/9, 192 = 1

Number of Image Nodes

Observed All:

100, 057 − 52, 217.71 ≈ 47, 839.29

Observed RP:

100, 057 − 52, 009.25 ≈ 48, 047.75

Maps

Theoretical:

(1 − e^{−1})n = (1 − e^{−1}) ∗ 100, 057 ≈ 63, 248.09

Observed None:

100, 057 − 74, 883.41 ≈ 25, 173.59

Observed Not PR:

100, 057 − 74, 835.10 ≈ 25, 221.90

Permutations

Theoretical:

n − 1 = 100, 057 − 1 = 100, 056

Observed PR:

100, 057 − 1 = 100, 056

Observed RPPR:

100, 057 − 1 = 100, 056

Average Tail Length

Observed All:

114.216

Observed RP:

114.86

Maps

Theoretical:

\sqrt{πn/8} = \sqrt{π100, 057/8} ≈ 198

Observed None:

113.94

Observed Not PR:

114.22

Permutations

Theoretical:

1

Observed PR:

1

Observed RPPR:

1

Average Cycle Length

Observed All:

115.32

Observed RP:

115.41

Maps

Theoretical:

\sqrt{πn/8} = \sqrt{π100, 057/8} ≈ 198

Observed None:

114.64

Observed Not PR:

114.67

Permutations

Theoretical:

\frac{n + 1}{2} = \frac{100, 057 + 1}{2} = 50, 029

Observed PR:

50, 191.4

Observed RPPR:

50, 454.6

Maximum Cycle Length

Observed: Observed All:

19, 028.01

Observed RP:

19, 218.92

Maps

Theoretical:

c_{1}\sqrt{n} ≈ 0.78248\sqrt{100, 057} ≈ 247.5

Observed None:

142.96

Observed Not PR:

143.02

Permutations

Theoretical:

0.62432965n = 0.62432965 ∗ 100057 ≈ 62, 468.55

Observed PR:

62, 627.74

Observed RPPR:

62, 896.84

Maximum Tail Length

Observed All:

218.15

Observed RP:

218.69

Maps

Theoretical:

c_{2}\sqrt{n} ≈ 1.73746\sqrt{100, 057} ≈ 549.59

Observed None:

311.53

Observed Not PR:

312.20

Permutations

Theoretical:

1

Observed PR:

1

Observed RPPR:

1