The l₂(p) and the λ(p)/l₂(p) ratio

of the first hundred million primes

where p an odd prime, l₂(p) the multiplicative order of 2 mod p and λ(p) the value of Carmichael function lamda at p

0. Introduction

This page contains:

The l₂(p) and the λ(p)/l₂(p) ratio of the first hundred million primes.
Statistics of the λ(p)/l₂(p) ratio (frequencies, counts, averages, maximum values).

1. The l₂(p) and the λ(p)/l₂(p) ratio of the first hundred million primes, in ranges of 1 million

1st million (except for 2)	2nd million	3rd million	4th million	5th million	6th million	7th million	8th million	9th million	10th million
11th million	12th million	13th million	14th million	15th million	16th million	17th million	18th million	19th million	20th million
21st million	22nd million	23rd million	24th million	25th million	26th million	27th million	28th million	29th million	30th million
31st million	32nd million	33rd million	34th million	35th million	36th million	37th million	38th million	39th million	40th million
41st million	42nd million	43rd million	44th million	45th million	46th million	47th million	48th million	49th million	50th million
51st million	52nd million	53rd million	54th million	55th million	56th million	57th million	58th million	59th million	60th million
61st million	62nd million	63rd million	64th million	65th million	66th million	67th million	68th million	69th million	70th million
71st million	72nd million	73rd million	74th million	75th million	76th million	77th million	78th million	79th million	80th million
81st million	82nd million	83rd million	84th million	85th million	86th million	87th million	88th million	89th million	90th million
91st million	92nd million	93rd million	94th million	95th million	96th million	97th million	98th million	99th million	100th million

2. Statistics of the λ(p)/l₂(p) ratio

2.1 The three values of the λ(p)/l₂(p) ratio with the higher frequencies

The three values of the λ(p)/l₂(p) ratio with the higher frequencies, both in each 1 million range and in the whole range of the first 125 million primes, are 1, 2 and 3.

In the following tables, the frequencies of these ratio values are being presented, for both the whole range of the first 125 million primes and for each 1 million range.

The frequencies of 1 are in the area of Artin's constant. The frequencies of the other ratio values are in other areas.

The frequencies of all λ(p)/l₂(p) ratio values for the first 125 million primes can be found here (xls, ods).

λ(p)/l₂(p)
in the range of the first 125 million primes

freq. %

1
46742468 37.3939746992%

2 35058208 28.0465666244%

3 8311077 6.6488616532%

λ(p)/l₂(p)
1st million 2nd million 3rd million 4th million 5th million 6th million 7th million 8th million 9th million 10th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
374022 37.4% 373801 37.38% 374121 37.41% 374071 37.41% 373910 37.39% 374024 37.4% 373946 37.39% 374084 37.41% 373962 37.4% 373908 37.39%

2 280793 28.08% 280250 28.02% 280288 28.03% 280943 28.09% 280252 28.03% 280126 28.01% 280888 28.09% 280232 28.02% 280532 28.05% 280582 28.06%

3 66464 6.65% 66307 6.63% 66486 6.65% 66405 6.64% 66593 6.66% 66485 6.65% 66501 6.65% 66364 6.64% 66533 6.65% 66343 6.63%

λ(p)/l₂(p)
11th million 12th million 13th million 14th million 15th million 16th million 17th million 18th million 19th million 20th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
374139 37.41% 374197 37.42% 374011 37.4% 373859 37.39% 373911 37.39% 373910 37.39% 373684 37.37% 374063 37.41% 374305 37.43% 374067 37.41%

2 279946 27.99% 280313 28.03% 280639 28.06% 280506 28.05% 280588 28.06% 280618 28.06% 280461 28.05% 280439 28.04% 280141 28.01% 280169 28.02%

3 66482 6.65% 66657 6.67% 66517 6.65% 66596 6.66% 66375 6.64% 66611 6.66% 66342 6.63% 66339 6.63% 66340 6.63% 66331 6.63%

λ(p)/l₂(p)
21st million 22nd million 23rd million 24th million 25th million 26th million 27th million 28th million 29th million 30th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373913 37.39% 373629 37.36% 373704 37.37% 374166 37.42% 374054 37.41% 374104 37.41% 373978 37.4% 373675 37.37% 374455 37.45% 373374 37.34%

2 280857 28.09% 279976 28.0% 280752 28.08% 279799 27.98% 281053 28.11% 280831 28.08% 280680 28.07% 280537 28.05% 280449 28.04% 280450 28.05%

3 66602 6.66% 66524 6.65% 66491 6.65% 66652 6.67% 66493 6.65% 66593 6.66% 66639 6.66% 66720 6.67% 66156 6.62% 66461 6.65%

λ(p)/l₂(p)
31st million 32nd million 33rd million 34th million 35th million 36th million 37th million 38th million 39th million 40th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373837 37.38% 374096 37.41% 373598 37.36% 374401 37.44% 373789 37.38% 373800 37.38% 373520 37.35% 374432 37.44% 373639 37.36% 374025 37.4%

2 280574 28.06% 280383 28.04% 280303 28.03% 279834 27.98% 280690 28.07% 280395 28.04% 280291 28.03% 280611 28.06% 280408 28.04% 280457 28.05%

3 66409 6.64% 66587 6.66% 66538 6.65% 66610 6.66% 66565 6.66% 66662 6.67% 66874 6.69% 66399 6.64% 66689 6.67% 66228 6.62%

λ(p)/l₂(p)
41st million 42nd million 43rd million 44th million 45th million 46th million 47th million 48th million 49th million 50th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
374146 37.41% 374269 37.43% 373537 37.35% 373880 37.39% 374429 37.44% 374017 37.4% 374010 37.4% 373933 37.39% 374451 37.45% 373837 37.38%

2 280949 28.09% 280697 28.07% 280137 28.01% 280756 28.08% 280309 28.03% 279747 27.97% 280576 28.06% 280734 28.07% 280201 28.02% 280772 28.08%

3 66646 6.66% 66133 6.61% 67083 6.71% 66174 6.62% 66207 6.62% 66687 6.67% 66115 6.61% 66603 6.66% 66280 6.63% 66750 6.67%

λ(p)/l₂(p)
51st million 52nd million 53rd million 54th million 55th million 56th million 57th million 58th million 59th million 60th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373083 37.31% 374004 37.4% 374347 37.43% 374408 37.44% 374179 37.42% 373983 37.4% 374041 37.4% 373956 37.4% 373477 37.35% 374239 37.42%

2 280873 28.09% 280480 28.05% 280533 28.05% 280591 28.06% 280160 28.02% 280607 28.06% 280740 28.07% 280580 28.06% 280716 28.07% 280618 28.06%

3 66692 6.67% 66244 6.62% 66446 6.64% 66565 6.66% 66758 6.68% 66512 6.65% 66289 6.63% 66460 6.65% 66412 6.64% 66434 6.64%

λ(p)/l₂(p)
61st million 62nd million 63rd million 64th million 65th million 66th million 67th million 68th million 69th million 70th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373583 37.36% 373480 37.35% 374031 37.4% 373587 37.36% 374003 37.4% 374446 37.44% 373930 37.39% 373852 37.39% 374095 37.41% 373482 37.35%

2 280423 28.04% 280516 28.05% 280236 28.02% 280463 28.05% 280278 28.03% 280333 28.03% 280402 28.04% 280490 28.05% 280328 28.03% 280694 28.07%

3 66798 6.68% 66460 6.65% 66755 6.68% 66700 6.67% 66630 6.66% 66275 6.63% 66088 6.61% 66723 6.67% 66171 6.62% 66636 6.66%

λ(p)/l₂(p)
71st million 72nd million 73rd million 74th million 75th million 76th million 77th million 78th million 79th million 80th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
374732 37.47% 373350 37.34% 373766 37.38% 373813 37.38% 373771 37.38% 373705 37.37% 373777 37.38% 374286 37.43% 373772 37.38% 374343 37.43%

2 280461 28.05% 280580 28.06% 280610 28.06% 280739 28.07% 280178 28.02% 280609 28.06% 280661 28.07% 280207 28.02% 280849 28.08% 280408 28.04%

3 66533 6.65% 66574 6.66% 66629 6.66% 66248 6.62% 66583 6.66% 66465 6.65% 66684 6.67% 66213 6.62% 66715 6.67% 66035 6.6%

λ(p)/l₂(p)
81st million 82nd million 83rd million 84th million 85th million 86th million 87th million 88th million 89th million 90th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
374089 37.41% 374360 37.44% 373785 37.38% 372847 37.28% 374045 37.4% 373171 37.32% 374057 37.41% 373540 37.35% 374297 37.43% 374098 37.41%

2 280374 28.04% 280822 28.08% 280058 28.01% 280673 28.07% 280294 28.03% 281069 28.11% 280383 28.04% 280175 28.02% 280315 28.03% 280550 28.05%

3 66630 6.66% 65949 6.59% 66559 6.66% 66779 6.68% 66277 6.63% 66984 6.7% 66281 6.63% 66656 6.67% 66373 6.64% 66424 6.64%

λ(p)/l₂(p)
91st million 92nd million 93rd million 94th million 95th million 96th million 97th million 98th million 99th million 100th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373817 37.38% 373736 37.37% 374334 37.43% 374655 37.47% 374100 37.41% 374164 37.42% 373771 37.38% 374292 37.43% 374038 37.4% 373841 37.38%

2 280508 28.05% 280203 28.02% 280374 28.04% 280499 28.05% 280378 28.04% 280701 28.07% 280341 28.03% 280549 28.05% 280322 28.03% 280644 28.06%

3 66341 6.63% 66372 6.64% 66200 6.62% 66349 6.63% 66580 6.66% 66427 6.64% 66519 6.65% 66267 6.63% 66430 6.64% 66657 6.67%

λ(p)/l₂(p)
101st million 102nd million 103rd million 104th million 105th million 106th million 107th million 108th million 109th million 110th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373855 37.39% 374295 37.43% 373929 37.39% 374178 37.42% 373671 37.37% 374041 37.4% 373870 37.39% 373583 37.36% 374177 37.42% 373764 37.38%

2 280354 28.04% 279719 27.97% 281032 28.1% 279613 27.96% 280885 28.09% 280108 28.01% 280431 28.04% 279820 27.98% 280726 28.07% 280551 28.06%

3 66808 6.68% 66255 6.63% 66386 6.64% 66460 6.65% 66702 6.67% 66420 6.64% 66938 6.69% 66499 6.65% 66553 6.66% 66212 6.62%

λ(p)/l₂(p)
111th million 112th million 113th million 114th million 115th million 116th million 117th million 118th million 119th million 120th million

freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. % freq. %

1
373907 37.39% 373481 37.35% 373904 37.39% 373971 37.4% 374037 37.4% 373327 37.33% 373802 37.38% 373628 37.36% 374199 37.42% 373863 37.39%

2 280215 28.02% 280044 28.0% 280441 28.04% 280580 28.06% 280053 28.01% 280394 28.04% 280353 28.04% 280939 28.09% 280695 28.07% 280979 28.1%

3 66205 6.62% 66855 6.69% 66305 6.63% 66514 6.65% 66701 6.67% 66333 6.63% 66712 6.67% 66267 6.63% 66740 6.67% 66400 6.64%

λ(p)/l₂(p)
121st million 122nd million 123rd million 124th million 125th million

freq. % freq. % freq. % freq. % freq. %

1
374196 37.42% 373740 37.37% 373998 37.4% 374144 37.41% 373657 37.37%

2 280599 28.06% 280872 28.09% 280086 28.01% 280791 28.08% 280419 28.04%

3 66294 6.63% 66379 6.64% 66695 6.67% 66408 6.64% 66614 6.66%

2.2 The number of the λ(p)/l₂(p) ratio values and the mean ratio value

In the following tables, the number of the λ(p)/l₂(p) ratio values, both in each 1 million range and in the whole range of the first 125 million primes, along with the new values appearing in each range and the mean ratio value, are being presented.

In the whole range of the first 125 million primes, there are 19512 ratio values.

The variation of the ratio values in each 1 million range is relatively small, ranging from a min of 1678 to a max of 1826.

The number of the new ratio values appearing in each 1 million range ranges from a min of 63 to a max of 731.

in the range of the first 125 million primes

Number of ratio values:
19512

Mean ratio value: 18.68

1st million 2nd million 3rd million 4th million 5th million 6th million 7th million 8th million 9th million 10th million

Number of ratio values in range:
1764
1755 1786 1744 1730 1784 1757 1723 1812 1792

Number of new ratio values in range:
1764 731 564 454 411 388 342 304 309 302

Mean ratio value: 12.78 14.32 15.01 13.95 13.82 16.0 18.33 14.43 16.32 18.35

11th million 12th million 13th million 14th million 15th million 16th million 17th million 18th million 19th million 20th million

Number of ratio values in range: 1757 1761 1740 1740 1741 1721 1793 1764 1753 1759

Number of new ratio values in range:
244 241 252 229 245 220 236 197 211 219

Mean ratio value: 18.42 20.56 16.28 13.51 18.32 14.57 16.78 16.94 14.6 18.08

21st million 22nd million 23rd million 24th million 25th million 26th million 27th million 28th million 29th million 30th million

Number of ratio values in range: 1826 1779 1777 1736 1762 1716 1783 1765 1741 1799

Number of new ratio values in range:
200 192 192 179 172 156 170 168 157 171

Mean ratio value: 16.3 16.21 21.32 14.81 14.69 15.93 16.56 18.43 24.49 17.73

31st million 32nd million 33rd million 34th million 35th million 36th million 37th million 38th million 39th million 40th million

Number of ratio values in range: 1740 1731 1762 1734 1756 1778 1747 1810 1759 1783

Number of new ratio values in range:
161 152 151 135 145 154 135 149 140 144

Mean ratio value: 19.01 16.17 31.55 20.68 15.48 15.66 13.86 31.02 22.21 17.67

41st million 42nd million 43rd million 44th million 45th million 46th million 47th million 48th million 49th million 50th million

Number of ratio values in range: 1799 1709 1762 1801 1781 1728 1772 1770 1744 1678

Number of new ratio values in range:
156 107 142 142 133 132 131 127 114 112

Mean ratio value: 18.57 16.07 19.57 21.67 17.27 16.05 14.19 13.74 14.57 13.88

51st million 52nd million 53rd million 54th million 55th million 56th million 57th million 58th million 59th million 60th million

Number of ratio values in range: 1804 1785 1739 1736 1727 1776 1776 1745 1758 1736

Number of new ratio values in range:
118 115 115 114 113 130 106 105 119 114

Mean ratio value: 15.01 16.95 14.15 14.72 14.77 16.65 15.16 15.22 19.47 20.75

61st million 62nd million 63rd million 64th million 65th million 66th million 67th million 68th million 69th million 70th million

Number of ratio values in range: 1772 1779 1787 1734 1796 1746 1808 1732 1710 1792

Number of new ratio values in range:
109 144 105 114 127 113 115 105 121 107

Mean ratio value: 18.87 17.77 17.95 14.07 16.19 15.89 24.97 21.07 15.1 23.98

71st million 72nd million 73rd million 74th million 75th million 76th million 77th million 78th million 79th million 80th million

Number of ratio values in range: 1683 1746 1720 1757 1814 1714 1718 1799 1756 1784

Number of new ratio values in range:
90 108 101 110 113 101 79 106 83 103

Mean ratio value: 14.43 19.89 15.16 19.98 19.62 15.07 17.89 16.2 14.43 24.81

81st million 82nd million 83rd million 84th million 85th million 86th million 87th million 88th million 89th million 90th million

Number of ratio values in range: 1779 1736 1751 1760 1730 1807 1731 1771 1728 1720

Number of new ratio values in range:
118 106 97 93 114 103 102 87 98 78

Mean ratio value: 24.24 19.94 15.14 18.44 14.83 15.12 17.37 20.96 15.18 14.04

91st million 92nd million 93rd million 94th million 95th million 96th million 97th million 98th million 99th million 100th million

Number of ratio values in range: 1755 1743 1768 1741 1768 1740 1775 1782 1710 1763

Number of new ratio values in range:
85 81 99 96 80 76 102 91 81 81

Mean ratio value: 30.36 15.57 27.47 15.87 16.08 18.88 17.77 16.67 23.42 16.99

101st million 102nd million 103rd million 104th million 105th million 106th million 107th million 108th million 109th million 110th million

Number of ratio values in range: 1776 1705 1745 1755 1755 1790 1765 1742 1752 1740

Number of new ratio values in range:
83 79 85 84 85 84 86 75 85 75

Mean ratio value: 17.1 26.69 14.65 18.3 15.64 87.01 17.57 15.02 14.08 21.1

111th million 112th million 113th million 114th million 115th million 116th million 117th million 118th million 119th million 120th million

Number of ratio values in range: 1752 1761 1763 1729 1800 1788 1784 1770 1743 1748

Number of new ratio values in range:
80 86 82 63 85 83 74 83 73 72

Mean ratio value: 15.3 13.93 15.93 13.61 16.11 16.4 40.31 16.13 17.94 15.96

121st million 122nd million 123rd million 124th million 125th million

Number of ratio values in range: 1778 1745 1777 1727 1743

Number of new ratio values in range:
64 68 77 79 69

Mean ratio value: 20.85 25.25 23.14 40.15 29.05

2.3 The maximum values of the λ(p)/l₂(p) ratio

The maximum values of the λ(p)/l₂(p) ratio and the prime p for which they occure are the following (Sequence A152598 in OEIS):

for p 3 (in the range of 1st million) new maximum ratio 1

for p 7 (in the range of 1st million) new maximum ratio 2

for p 31 (in the range of 1st million) new maximum ratio 6

for p 73 (in the range of 1st million) new maximum ratio 8

for p 127 (in the range of 1st million) new maximum ratio 18

for p 601 (in the range of 1st million) new maximum ratio 24

for p 683 (in the range of 1st million) new maximum ratio 31

for p 1103 (in the range of 1st million) new maximum ratio 38

for p 1801 (in the range of 1st million) new maximum ratio 72

for p 2731 (in the range of 1st million) new maximum ratio 105

for p 5419 (in the range of 1st million) new maximum ratio 129

for p 8191 (in the range of 1st million) new maximum ratio 630

for p 43691 (in the range of 1st million) new maximum ratio 1285

for p 61681 (in the range of 1st million) new maximum ratio 1542

for p 65537 (in the range of 1st million) new maximum ratio 2048

for p 121369 (in the range of 1st million) new maximum ratio 3112

for p 122921 (in the range of 1st million) new maximum ratio 3512

for p 131071 (in the range of 1st million) new maximum ratio 7710

for p 178481 (in the range of 1st million) new maximum ratio 7760

for p 262657 (in the range of 1st million) new maximum ratio 9728

for p 524287 (in the range of 1st million) new maximum ratio 27594

for p 2099863 (in the range of 1st million) new maximum ratio 48834

for p 2796203 (in the range of 1st million) new maximum ratio 60787

for p 6700417 (in the range of 1st million) new maximum ratio 104694

for p 10567201 (in the range of 1st million) new maximum ratio 140896

for p 13264529 (in the range of 1st million) new maximum ratio 282224

for p 20394401 (in the range of 2nd million) new maximum ratio 384800

for p 48544121 (in the range of 3rd million) new maximum ratio 683720

for p 97685839 (in the range of 6th million) new maximum ratio 1205998

for p 112901153 (in the range of 7th million) new maximum ratio 1240672

for p 160465489 (in the range of 9th million) new maximum ratio 1407592

for p 164511353 (in the range of 10th million) new maximum ratio 4012472

for p 420778751 (in the range of 23rd million) new maximum ratio 4429250

for p 536903681 (in the range of 29th million) new maximum ratio 4628480

for p 616318177 (in the range of 33rd million) new maximum ratio 16657248

for p 2147483647 (in the range of 106th million) new maximum ratio 69273666

Multiplicative Order Calculator

Last updated on 13/03/2010 by V. Papadimitriou (bpapa at sch dot gr)

λ(p)/l₂(p)	in the range of the first 125 million primes
λ(p)/l₂(p)	freq.	%
1	46742468	37.3939746992%
2	35058208	28.0465666244%
3	8311077	6.6488616532%

λ(p)/l₂(p)	1st million		2nd million		3rd million		4th million		5th million		6th million		7th million		8th million		9th million		10th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374022	37.4%	373801	37.38%	374121	37.41%	374071	37.41%	373910	37.39%	374024	37.4%	373946	37.39%	374084	37.41%	373962	37.4%	373908	37.39%
2	280793	28.08%	280250	28.02%	280288	28.03%	280943	28.09%	280252	28.03%	280126	28.01%	280888	28.09%	280232	28.02%	280532	28.05%	280582	28.06%
3	66464	6.65%	66307	6.63%	66486	6.65%	66405	6.64%	66593	6.66%	66485	6.65%	66501	6.65%	66364	6.64%	66533	6.65%	66343	6.63%

λ(p)/l₂(p)	11th million		12th million		13th million		14th million		15th million		16th million		17th million		18th million		19th million		20th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374139	37.41%	374197	37.42%	374011	37.4%	373859	37.39%	373911	37.39%	373910	37.39%	373684	37.37%	374063	37.41%	374305	37.43%	374067	37.41%
2	279946	27.99%	280313	28.03%	280639	28.06%	280506	28.05%	280588	28.06%	280618	28.06%	280461	28.05%	280439	28.04%	280141	28.01%	280169	28.02%
3	66482	6.65%	66657	6.67%	66517	6.65%	66596	6.66%	66375	6.64%	66611	6.66%	66342	6.63%	66339	6.63%	66340	6.63%	66331	6.63%

λ(p)/l₂(p)	21st million		22nd million		23rd million		24th million		25th million		26th million		27th million		28th million		29th million		30th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373913	37.39%	373629	37.36%	373704	37.37%	374166	37.42%	374054	37.41%	374104	37.41%	373978	37.4%	373675	37.37%	374455	37.45%	373374	37.34%
2	280857	28.09%	279976	28.0%	280752	28.08%	279799	27.98%	281053	28.11%	280831	28.08%	280680	28.07%	280537	28.05%	280449	28.04%	280450	28.05%
3	66602	6.66%	66524	6.65%	66491	6.65%	66652	6.67%	66493	6.65%	66593	6.66%	66639	6.66%	66720	6.67%	66156	6.62%	66461	6.65%

λ(p)/l₂(p)	31st million		32nd million		33rd million		34th million		35th million		36th million		37th million		38th million		39th million		40th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373837	37.38%	374096	37.41%	373598	37.36%	374401	37.44%	373789	37.38%	373800	37.38%	373520	37.35%	374432	37.44%	373639	37.36%	374025	37.4%
2	280574	28.06%	280383	28.04%	280303	28.03%	279834	27.98%	280690	28.07%	280395	28.04%	280291	28.03%	280611	28.06%	280408	28.04%	280457	28.05%
3	66409	6.64%	66587	6.66%	66538	6.65%	66610	6.66%	66565	6.66%	66662	6.67%	66874	6.69%	66399	6.64%	66689	6.67%	66228	6.62%

λ(p)/l₂(p)	41st million		42nd million		43rd million		44th million		45th million		46th million		47th million		48th million		49th million		50th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374146	37.41%	374269	37.43%	373537	37.35%	373880	37.39%	374429	37.44%	374017	37.4%	374010	37.4%	373933	37.39%	374451	37.45%	373837	37.38%
2	280949	28.09%	280697	28.07%	280137	28.01%	280756	28.08%	280309	28.03%	279747	27.97%	280576	28.06%	280734	28.07%	280201	28.02%	280772	28.08%
3	66646	6.66%	66133	6.61%	67083	6.71%	66174	6.62%	66207	6.62%	66687	6.67%	66115	6.61%	66603	6.66%	66280	6.63%	66750	6.67%

λ(p)/l₂(p)	51st million		52nd million		53rd million		54th million		55th million		56th million		57th million		58th million		59th million		60th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373083	37.31%	374004	37.4%	374347	37.43%	374408	37.44%	374179	37.42%	373983	37.4%	374041	37.4%	373956	37.4%	373477	37.35%	374239	37.42%
2	280873	28.09%	280480	28.05%	280533	28.05%	280591	28.06%	280160	28.02%	280607	28.06%	280740	28.07%	280580	28.06%	280716	28.07%	280618	28.06%
3	66692	6.67%	66244	6.62%	66446	6.64%	66565	6.66%	66758	6.68%	66512	6.65%	66289	6.63%	66460	6.65%	66412	6.64%	66434	6.64%

λ(p)/l₂(p)	61st million		62nd million		63rd million		64th million		65th million		66th million		67th million		68th million		69th million		70th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373583	37.36%	373480	37.35%	374031	37.4%	373587	37.36%	374003	37.4%	374446	37.44%	373930	37.39%	373852	37.39%	374095	37.41%	373482	37.35%
2	280423	28.04%	280516	28.05%	280236	28.02%	280463	28.05%	280278	28.03%	280333	28.03%	280402	28.04%	280490	28.05%	280328	28.03%	280694	28.07%
3	66798	6.68%	66460	6.65%	66755	6.68%	66700	6.67%	66630	6.66%	66275	6.63%	66088	6.61%	66723	6.67%	66171	6.62%	66636	6.66%

λ(p)/l₂(p)	71st million		72nd million		73rd million		74th million		75th million		76th million		77th million		78th million		79th million		80th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374732	37.47%	373350	37.34%	373766	37.38%	373813	37.38%	373771	37.38%	373705	37.37%	373777	37.38%	374286	37.43%	373772	37.38%	374343	37.43%
2	280461	28.05%	280580	28.06%	280610	28.06%	280739	28.07%	280178	28.02%	280609	28.06%	280661	28.07%	280207	28.02%	280849	28.08%	280408	28.04%
3	66533	6.65%	66574	6.66%	66629	6.66%	66248	6.62%	66583	6.66%	66465	6.65%	66684	6.67%	66213	6.62%	66715	6.67%	66035	6.6%

λ(p)/l₂(p)	81st million		82nd million		83rd million		84th million		85th million		86th million		87th million		88th million		89th million		90th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374089	37.41%	374360	37.44%	373785	37.38%	372847	37.28%	374045	37.4%	373171	37.32%	374057	37.41%	373540	37.35%	374297	37.43%	374098	37.41%
2	280374	28.04%	280822	28.08%	280058	28.01%	280673	28.07%	280294	28.03%	281069	28.11%	280383	28.04%	280175	28.02%	280315	28.03%	280550	28.05%
3	66630	6.66%	65949	6.59%	66559	6.66%	66779	6.68%	66277	6.63%	66984	6.7%	66281	6.63%	66656	6.67%	66373	6.64%	66424	6.64%

λ(p)/l₂(p)	91st million		92nd million		93rd million		94th million		95th million		96th million		97th million		98th million		99th million		100th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373817	37.38%	373736	37.37%	374334	37.43%	374655	37.47%	374100	37.41%	374164	37.42%	373771	37.38%	374292	37.43%	374038	37.4%	373841	37.38%
2	280508	28.05%	280203	28.02%	280374	28.04%	280499	28.05%	280378	28.04%	280701	28.07%	280341	28.03%	280549	28.05%	280322	28.03%	280644	28.06%
3	66341	6.63%	66372	6.64%	66200	6.62%	66349	6.63%	66580	6.66%	66427	6.64%	66519	6.65%	66267	6.63%	66430	6.64%	66657	6.67%

λ(p)/l₂(p)	101st million		102nd million		103rd million		104th million		105th million		106th million		107th million		108th million		109th million		110th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373855	37.39%	374295	37.43%	373929	37.39%	374178	37.42%	373671	37.37%	374041	37.4%	373870	37.39%	373583	37.36%	374177	37.42%	373764	37.38%
2	280354	28.04%	279719	27.97%	281032	28.1%	279613	27.96%	280885	28.09%	280108	28.01%	280431	28.04%	279820	27.98%	280726	28.07%	280551	28.06%
3	66808	6.68%	66255	6.63%	66386	6.64%	66460	6.65%	66702	6.67%	66420	6.64%	66938	6.69%	66499	6.65%	66553	6.66%	66212	6.62%

λ(p)/l₂(p)	111th million		112th million		113th million		114th million		115th million		116th million		117th million		118th million		119th million		120th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	373907	37.39%	373481	37.35%	373904	37.39%	373971	37.4%	374037	37.4%	373327	37.33%	373802	37.38%	373628	37.36%	374199	37.42%	373863	37.39%
2	280215	28.02%	280044	28.0%	280441	28.04%	280580	28.06%	280053	28.01%	280394	28.04%	280353	28.04%	280939	28.09%	280695	28.07%	280979	28.1%
3	66205	6.62%	66855	6.69%	66305	6.63%	66514	6.65%	66701	6.67%	66333	6.63%	66712	6.67%	66267	6.63%	66740	6.67%	66400	6.64%

λ(p)/l₂(p)	121st million		122nd million		123rd million		124th million		125th million
λ(p)/l₂(p)	freq.	%	freq.	%	freq.	%	freq.	%	freq.	%
1	374196	37.42%	373740	37.37%	373998	37.4%	374144	37.41%	373657	37.37%
2	280599	28.06%	280872	28.09%	280086	28.01%	280791	28.08%	280419	28.04%
3	66294	6.63%	66379	6.64%	66695	6.67%	66408	6.64%	66614	6.66%

	in the range of the first 125 million primes
Number of ratio values:	19512
Mean ratio value:	18.68

for p 3	(in the range of 1st million)	new maximum ratio 1
for p 7	(in the range of 1st million)	new maximum ratio 2
for p 31	(in the range of 1st million)	new maximum ratio 6
for p 73	(in the range of 1st million)	new maximum ratio 8
for p 127	(in the range of 1st million)	new maximum ratio 18
for p 601	(in the range of 1st million)	new maximum ratio 24
for p 683	(in the range of 1st million)	new maximum ratio 31
for p 1103	(in the range of 1st million)	new maximum ratio 38
for p 1801	(in the range of 1st million)	new maximum ratio 72
for p 2731	(in the range of 1st million)	new maximum ratio 105
for p 5419	(in the range of 1st million)	new maximum ratio 129
for p 8191	(in the range of 1st million)	new maximum ratio 630
for p 43691	(in the range of 1st million)	new maximum ratio 1285
for p 61681	(in the range of 1st million)	new maximum ratio 1542
for p 65537	(in the range of 1st million)	new maximum ratio 2048
for p 121369	(in the range of 1st million)	new maximum ratio 3112
for p 122921	(in the range of 1st million)	new maximum ratio 3512
for p 131071	(in the range of 1st million)	new maximum ratio 7710
for p 178481	(in the range of 1st million)	new maximum ratio 7760
for p 262657	(in the range of 1st million)	new maximum ratio 9728
for p 524287	(in the range of 1st million)	new maximum ratio 27594
for p 2099863	(in the range of 1st million)	new maximum ratio 48834
for p 2796203	(in the range of 1st million)	new maximum ratio 60787
for p 6700417	(in the range of 1st million)	new maximum ratio 104694
for p 10567201	(in the range of 1st million)	new maximum ratio 140896
for p 13264529	(in the range of 1st million)	new maximum ratio 282224
for p 20394401	(in the range of 2nd million)	new maximum ratio 384800
for p 48544121	(in the range of 3rd million)	new maximum ratio 683720
for p 97685839	(in the range of 6th million)	new maximum ratio 1205998
for p 112901153	(in the range of 7th million)	new maximum ratio 1240672
for p 160465489	(in the range of 9th million)	new maximum ratio 1407592
for p 164511353	(in the range of 10th million)	new maximum ratio 4012472
for p 420778751	(in the range of 23rd million)	new maximum ratio 4429250
for p 536903681	(in the range of 29th million)	new maximum ratio 4628480
for p 616318177	(in the range of 33rd million)	new maximum ratio 16657248
for p 2147483647	(in the range of 106th million)	new maximum ratio 69273666

The l2(p) and the λ(p)/l2(p) ratio

of the first hundred million primes

where p an odd prime, l2(p) the multiplicative order of 2 mod p and λ(p) the value of Carmichael function lamda at p

0. Introduction

1. The l2(p) and the λ(p)/l2(p) ratio of the first hundred million primes, in ranges of 1 million

2. Statistics of the λ(p)/l2(p) ratio

2.1 The three values of the λ(p)/l2(p) ratio with the higher frequencies

2.2 The number of the λ(p)/l2(p) ratio values and the mean ratio value