Algebraic Expansion Of The Collatz Sequence
Bradley Berg December 30, 2022
Updated: June 8, 2025
Notation: #3a0d Hexadecimal values have a leading pound sign.
X*Y Multiplication
X^Y Exponentiation
log2( x ) Log base 2 of X
The original Collatz sequence is:
N is Even: N' = N / 2
N is Odd: N' = 3 * N + 1
A variation is to rewrite it as a series of Odd steps. In this form Ei is the number of Even steps in each transition. This is also the number of trailing zeros in the result from each Odd transition. It will always be one or more. The second Odd form runs a little faster and easier to code when computing sequence values.
N0 = Seed
Ni * 3 + 1 N + (N + 1) / 2
Ni+1 = ----------- = ---------------
2^Ei 2^Ei
To misappropriate the terms from astrophysics, the point where the sequence dips below the starting Seed is the (event) horizon. Any series that goes below the horizon collapses to the singularity, one.
Here a run length is defined as the number of odd steps until the sequence goes below the horizon. This definition is useful because it corresponds wo the number of terms in the algebraic expansion.
The algebraic expansion of a full run of length, L, is:
Seed * 3^L + DL
NL = ---------------
2^EL
L
where: DL = ∑ 3^(L-j) * 2^Kj-1
j=1
Kj-1 is the total number of even transitions up through step j. To expand this sequence it is useful to define Kj-1 as a sequence.
K0 = 0 Ki = Ki-1 + Ei
This lets us expand the sequence to form an iterative sequence.
N0 = Seed Only odd Seeds are allowed
N1 = Seed * 3 + 1 / 2^K1 K1 = E1
N2 = Seed * 3 + 1 Seed*9 + 3 + 2^K1
------------ * 3 + 1 = -----------------
2^K1 2^K2
----------------
2^E2
N3 = Seed*9 + 3 + 2^K1 Seed*27 + 9 + 3*2^K1 + 2^K2
----------------- * 3 + 1 = ----------------------------
2^K2 2^K3
-----------------
2^E3
i
Ni = Seed*3^i + ∑ 3^(i-j) * 2^Kj-1
j=1
--------------------------------
2^Ki
To simplify the expansion let Di signify the sum. This term is useful since Di encapsulates the non-algebraic portion of the sequence known as the hailstone effect.
Seed * 3^i + Di i
Ni = --------------- Di = ∑ 3^(i-j) * 2^Kj-1
2^Ki j=1
Di can also be computed using a recursive sequence derived directly from the Collatz sequence. This recursive form is useful to validate that Di is properly calculated.
D0 = 0 Di+1 = Di * 3 + 2^Ki
The value of KL is derived from the final value in a run.
Seed * 3^L + DL
NL = --------------- < Seed
2^KL
Since the values of Ni only drop below the Seed at the end of a run then 2^KL will be just large enough to satisfy this constrint. Dividing both sides by the Seed we get:
3^L + DL / Seed
---------------- < 1
2^KL
Here the term
3^L < 2^KL and KL = ceiling{ log2( 3^L )}
This example combines all four sequence components; showing how they interrelate. The horizon line is the point where the series drops below the Seed.
i Ei Ki Ni Di 0 0 0 139 0 Seed = 139 1 1 1 209 1 (139*3^1 + 3^0*2^0) / 2^1 2 2 3 157 5 (139*3^2 + 3^1*2^0 + 3^0*2^1) / 2^3 -------------------------------------------- Horizon 3 3 6 59 23 (139*3^3 + 3^2*2^0 + 3^1*2^1 + 3^0*2^3) / 2^6 4 1 7 89 133 5 2 9 67 527 6 1 10 101 2093 7 4 14 19 7303 8 1 15 29 38293 9 3 18 11 147647 10 1 19 17 705085 11 2 21 13 2639543 12 3 24 5 10015781 13 4 28 1 46824559
Here is the algebraic expansion for a longer run with a Seed of 359.
Ni * 2^Ki = Seed * 3^i + Di
i 1 2 3 4 5 6 7 8 9 10
Ei 1 1 2 1 1 1 1 4 2 2
Ki 1 2 4 5 6 7 8 12 14 16
Ni 539 809 607 911 1367 2051 3077 577 433 325
Ni*2^Ki 1078 3236 9712 29152 87488 262528 787712 2363392 7094272 21299200
Seed*3^i 359*3^1 359*3^2 359*3^3 359*3^4 359*3^5 359*3^6 359*3^7 359*3^8 359*3^9 359*3^10
Di 1 5 19 73 251 817 2579 7993 28075 100609
Di 3^0*2^0 3^1*2^0 3^2*2^0 3^3*2^0 3^4*2^0 3^5*2^0 3^6*2^0 3^7*2^0 3^8*2^0 3^9*2^0
3^0*2^1 3^1*2^1 3^2*2^1 3^3*2^1 3^4*2^1 3^5*2^1 3^6*2^1 3^7*2^1 3^8*2^1
3^0*2^2 3^1*2^2 3^2*2^2 3^3*2^2 3^4*2^2 3^5*2^2 3^6*2^2 3^7*2^2
3^0*2^4 3^1*2^4 3^2*2^4 3^3*2^4 3^4*2^4 3^5*2^4 3^6*2^4
3^0*2^5 3^1*2^5 3^2*2^5 3^3*2^5 3^4*2^5 3^5*2^5
3^0*2^6 3^1*2^6 3^2*2^6 3^3*2^6 3^4*2^6
3^0*2^7 3^1*2^7 3^2*2^7 3^3*2^7
3^0*2^8 3^1*2^8 3^2*2^8
3^0*2^12 3^1*2^12
3^0*2^14
In this section we derive these upper and lower bounds on the DL term.
3^L - 2^L <= DL < L * 3^(L - 1)
The sum DL depends on the values Ki; which are the total number of even steps following each odd step. Conway[1] showed they cannot be determined algebraically due to thier randomized behavior; creating the hailstone effect. However, we can derive bounds on DL for a sequence with L Odd steps.
L
DL = ∑ 3^(L-j) * 2^Kj-1
j=1
As the terms progress the powers of 3 are decremented in each step while the powers of two increase. Each of these values balance out so that all the terms are of the same magnitute. A balance is maintained because the powers of two are limited by the Ki. When Ki value gets too big then the corresponding value in the run dips below the horizon and terminates.
A maximum bound for DL is determined setting Ki values so they produce a large value. This is done by setting them to their maximum value early in the run. Since the powers of 3 are biggest early in the run, these terms get even larger with larger Ki values.
As the series progresses the Ki values are the number of
even steps and the index, i, is the number of odd steps. To stay above the
horizon then 3^j must be greater than
E = 0; Starting exponent for 2^Ei K = 0; Starting sum of exponent values Dmax = 3^(L-1); Power of 3 in the first term of DL DO J = 2 to L: DO over terms up to the run length, E = [log2( 3^(J-1) )] - K; Maximum possible exponent K += E; Sum of exponents Dmax += 3^(L-J) * 2^K; Sum of allowed maximum terms -
The first term of the series,
3^j > 2^Kj-1
3^(L-j) * 3^j > 3^(L-j) * 2^Kj-1 previous > current term
3^(L-1) > 3^(L-j) * 2^Kj-1 first term > any other term
L
DL = ∑ 3^(L-j) * 2^Kj-1 < L * 3^(L-1)
j=1
The terms of the sum are close in magnitude so
K = 0; Starting sum of exponent values Dmin = 3^(L-1); Power of 3 in the first term of DL DO j = 2 to L: DO over terms up to the run length, K += 1; Sum of exponents equal to one Dmin += 3^(L-j) * 2^K; Sum of minimum terms -
The lower bound produced by the algorithm can be calculated exactly
as the sum of two parts.
The first term,
Dmin = 3^(L-1) Initial term
+ 3 * 2^(L-1) * (1.5^(L-2) - 1) Geometric term
The sum of the remaining terms are evaluated in reverse order.
The last term is
T = 2^(L-1) Last term
Sum = T*1.5 + T*1.5^2 + T*1.5^3 + ... + T*1.5^(L-1)
Sum = T * {1 + 1.5 + 1.5^2 + 1.5^3 + ... + T*1.5^(L-1)}
The sum of the geometric series yield the lower bound on DL.
L-1 1 - 1.5^(L-1) 1 - 1.5^(L-1)
∑ 1.5^i = ------------- = ------------- = 1.5^(L-1) * 2
i=0 1 - 1.5 -.5
Sum = T * 1.5^(L-1) * 2
= 2^(L-1) * 1.5^(L-1) * 2
= { 3^(L-1) - 2^(L-1) } * 2
= 2 * 3^(L-1) - 2^L
DL >= 3^(L-1) + 2 * 3^(L-1) - 2^L
DL >= 3^L - 2^L
Together the upper and lower bounds are:
3^L - 2^L <= DL < L * 3^(L-1)
For verification these charts list bounds after running many Seeds that
produced runs of a given length. The bounds were validated
beyond this chart for Seeds up to
L * 3^(L-1) Observed
L Maximum Bound Maximum DL
10 #300de #169c9
15 #446bc17 #33e4817
20 #5_69860d1c #4_069e6dd5
25 #66b_f4ce0df9 #499_7c31d84f
29 #25b62_218c688d #19582_9cf3badf
30 #75091_a5e8b73a #4b724_33ff27fd
35 #819b94b_3b36e8bb #50c4ef4_133b9527
40 #8_c99eb99c_cefec1b8 #5_33af2189_7da9a539
41 #1b_0594e128_961c2d49 #f_b49c4be8_03b5a2cf
45 #962_4ddf75d3_8b4c1ddd #59d_d56ec09e_ed78837f
50 #9e5ae_21ae451c_ea477f16 #5edd6_78b6ac73_a7ff33a1
55 #a5584b7_c4765d17_6ba6fedf #563376f_74f8d091_54e81f4b
60 #a_b376e2a8_2a911fe3_79a731d4 #5_70b66f89_7c324334_70545e89
3^L - 2^L Observed
L Minimum Bound Minimum DL
10 #e2a9 #18901
15 #da726b #1f603b7
20 #cfc41b91 #1_f29d7af1
25 #c5_44562aa3 #28e_69d42f4f
29 #3e6b_21437d93 #b088_507802db
30 #bb41_83ca78b9 #2b9d1_6ab71fb1
35 #b1bf64_d930979b #1a4445c_586dabe3
40 #a8b8b352_291fe821 #e28c7dbb_ac598c21
41 #1_fa2a1af6_7b5fb863 #1_ffdee34b_9a06b863
45 #a0_275309fd_09495753 #a0_baeb3fce_b4355753
50 #9805_53ecdb2f_d09de3c9 #982B_32dba1a9_a4dde3c9
55 #904d0e_ad200e63_05df37cb #911646_4dcfb2c2_6c9f37cb
60 #88f924ee_beeda7fe_92e1f5b1 #8a3b121e_a0239504_0bfef5b1
The majority of runs are trivialy short. The length of a short run can easily be determinted using the low order bits of the Seed. When the low bits for different Seeds match, if the run length is short it will be the same regardless of the upper bits. However, in non-trivial runs the low order bits can not be used to determine the run length.
Starting with the low 4 bits of a Seed we can tell the the length of many runs other than 7, #B, or #F. The short lengths are either 1 or 2.
Low bits: 1 3 5 7 9 B D F Length: 1 2 1 ? 1 ? 1 ?
By constructing an array with entries of the non-trivial low bits you can generate Seeds that only have longer runs. With the example above 5 out of 8 consecutive runs will be trivial. Note that if the Seed is even it is immediately divided by 2 and goes below the Seed. Consequnetly even Seeds can be discarded out of hand.
The reason that short runs all have the same lower bits is that the
first few terms of DL will be computed the same way.
For trivial runs the corresponding run will drop below the horizon before
any of the upper bits in the Seed can have an effect.
Using the earlier example where the Seed is
Here is a pyramid with low order bits (in hexadecimal) for non-trivial runs ranging from 1 to 8 bits. Each branch to the left adds a one bit to the front of a Seed. Right branches adds a leading zero. The leaf nodes at the bottom are the low 8 bits of Seeds needed to for a long run. Here only 19 out of 128 odd candidate Seeds produce runs over 3 steps.
1
|
____________ 3 ____________
/ \
___________ 7 ___________ 3
/ \ |
_________ f _________ 7 b
/ \ | |
___ 1f ___ _ f _ _ 7 _ 1b
/ \ / \ / \ / \
3f 1f 2f f 27 7 3b 1b
/ \ / \ / \ | / \ | / \ |
7f 3f 5f 1f 6f 2f 4f 67 27 47 7b 5b 1b
/ \ / \ | / \ / \ | | / \ | | / \ | |
ff 7f bf 3f df 9f 1f ef 6f 2f cf e7 67 a7 47 fb 9b 5b 1b
To produce a series of candidate Seeds that produce long runs you can combine low Seed bits with upper bits of your choosing. The low bits would be in the set of values for non-trivial runs as listed above.
An array of 20 bit low order values for long runs has 27328 elements. Using it to create Seeds eliminates 94.8% of values that are trivial. This was used to speed up exhaustive searches in experimental runs. Since only short runs are eliminated the performance gain is not as much as it might seem, but still there is no cost. To generate Seeds without trivial runs use nested loops as below using an array of low bit values named Long.Run.
DO Upper = 0 to Limit: DO over Seeds (may start and end anywhere),
DO @Lower in Long.Run: DO over the array of the lower 20 bits,
Seed = (Upper * 2^20) + Lower; Seed with a non-trivial length
< Process the Seed here as you wish. >
- -
An interesting observation is that any one of these entries seem to produce Seeds that cover all the same run lengths as any other entry. Using Seeds up to 49 bits wide it was verified that all entries produce all lengths up to 182 Odd steps. This makes sense since the values in the run are randomized so they eventually stumble into a given length.
Appendix A has a program that generates the array of low order 20 bit values. For a single value in the array, Appendix B has a method that extends candidate low order values from 20 to 36 bits.
The lowest Seed for runs with up to 471 Odd steps were found. This was sped up by filtering out Seeds with in trivial runs. Appendix C lists the lowest Seeds for runs up to 471 steps. It also includes Seeds for some longer runs, but also adds some that are larger than the lowest Seed.
This chart puts the run length on the horizontal access and
Lengths below 50 are excluded as that is the crossover point between the maximum value of DL and its average value. For smaller lengths the maximum value dominates and those values are characterized by smaller than average Seed values for longer runs.
Trendline for the log2( Seed ) 0.0865 * L + 10.86 Largest known variance for a Seed 1_00893_22492_96231 @ L = 559 Estimate for log2( Seed ) at 559 59.2135 Actual value of log2( Seed ) 49.8418 Lower bound on all known Seeds 3 * 1.061^L
Despite substantial variance the trend appears to be linear. The sample size is still too small to draw any conclusion beyond that. Computing additional longer runs quickly becomes prohibitively difficult.
The seven longest runs were found by running Seeds up through 53 bits.
Length = 551 Seed = 5_74618_44510_69947 = #14_6a1f_d305_17fb Length = 552 Seed = 3_83078_96340_46631 = #d_9c15_3758_baa7 Length = 553 Seed = 5_10771_95120_62175 = #12_2571_9f20_f8df Length = 555 Seed = 2_55385_97560_31087 = #9_12b8_cf90_7c6f Length = 556 Seed = 1_70257_31706_87391 = #6_0c7b_350a_fd9f Length = 558 Seed = 1_51339_83739_44347 = #5_606d_847b_8c1b Length = 559 Seed = 1_00893_22492_96231 = #3_959e_5852_5d67
This distribution chart divides long runs into groups of fifty. There are not enough samples to draw any conclusions other than they are very sparsly distributed. Extending this table beyond 53 bits might help, but requires a lot of compute power.
-- Number of Run Lengths --
Bits 350 400 450 500 550
in to to to to to
Seeds 399 450 499 549 599
44 23 2 0 0 0
45 59 6 0 0 0
46 125 14 0 0 0
47 232 19 1 0 0
48 503 27 0 0 0
49 948 47 4 0 0
50 many 96 3 0 1
51 many 63 10 0 3
52 many 28 22 0 3
53 many 35 52 6 2
To see what happens with larger seeds we ran only seeds with many low order one bits up through 64 bits. The largest runs found were:
Length = 521 Ones = 22 Seed = 15436_52028_83233_71007 = #d639_896a_819f_ffff Length = 499 Ones = 23 Seed = 4198_10985_60470_38463 = #3a42_aeab_e9bf_ffff Length = 500 Ones = 24 Seed = 2798_73990_40313_58975 = #26d7_1f1d_467f_ffff Length = 500 Ones = 25 Seed = 5597_47980_80627_17951 = #4dae_3e3a_8cff_ffff
To see how 64 bit seeds performed, Seeds were run from #8000_1000_0000_0000 to #8004_0000_0000_0000. The three largest runs found were previously found with smaller Seeds.
Length = 475 Seed = 9224_03530_70197_42527 = #8002_5b3d_9e92_ad3f Length = 499 Seed = 9223_73915_13254_56383 = #8001_4de3_7f9d_e7ff Length = 555 Seed = 9224_23687_00004_42011 = #8003_128f_a86f_fa9b
Initially, as the seed gets wider ever larger run lengths are found. Also, there are fewer longer lengths as there are far more shorter lengths. It could be that they've just become very scarce and hard to find.
Another possibility is that there is a limit to run lengths. The total number of bits in a run increases by its length per additional bit in the Seed. Bits are randomized as the total number of bits in a run increases. In turn entropy also increases and long runs trend towards more average behaviour. Increased entropy in a run makes it reach the horizon in fewer steps proportional to its size. Despite using very large seeds, at some point they may not produce larger runs.
So far we only have runs up through 53 bit Seeds. It would take running substantially larger Seeds to even get a glimpse at the distribution of long run lengths. Even then further results would not be able to demonstrate a conclusion, but might give a sense of the overall behaviour.
[1] John H. Conway, "Unpredictable iterations". Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49-52.
[2] Techneon, "Gilda Language Reference", techneon.com/gilda/reference/
There are only 27328 Seeds out of a possible
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:
method LONG: Print the low 20 bits of the 27328 Seeds resulting in long runs.
local Seed cell, &Extended Seed
First word, &Initial run length
Length word :Subsequent run length
:
:...............................................................................
DO Low = 3 to #f_ffff by 2: DO over odd 20 bit numbers,
RUN.LENGTH Low, First; Run length of a 20 bit Seed
DO Hi = 1 to 3: DO over two extended bits,
Seed = [Hi \\ 20] \/ Low; Prepend the upper bits.
RUN.LENGTH Seed, Length; Run length of an extended Seed
IF Length ~= First: IF the run lengths are not the same,
PRINT Low; Print a long 20 bit Seed.
UNDO; UNDO
- - .
return
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:
idiom RUN.LENGTH: Number of steps in a run of the Collatz sequence
entry S cell :Seed for the run (must be odd)
exit L word :Length of the run
local N cell :Sequence values
:
:...............................................................................
N = S; Start of the sequence
DO until N < S: DO until the sequence runs below the Seed,
N = (3 * N + 1) / 2; Take an odd step.
L += 1; Count only odd steps.
DO until N /\ 1: DO until the next odd value,
N /= 2; Take an even step.
- -
return
The filter based on the low 20 bits can be extended by creating a secondary array. For each 20 bit low value we can compute the next higher 16 bits. This results in a set of 36 bit low bits that eliminates an additional 75% of candidate long runs. This subroutine witten in Gilda[2] implements the algorithm.
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:
method UPPER16: Create a table of 36 bit long seeds.
entry Low word :Low 20 bits for a long run
exit Long[#ffff] parcel, &Array of next 16 bits
Count word :Number of array entries
local Seed cell,
Base cell,
High cell,
Length word
:
:...............................................................................
: Create a table of 36 bit long seeds.
:
DO Mid = 0 to 2^16-1: DO over 16 bits
Base = [Mid \\ 20] \/ Low
Length = Length.Seed( Base )
DO High = 1 to 6: DO only loop to 6!
Seed = [High \\ 36] \/ Base
IF Length ~= Length.Seed( Seed ): IF long base,
Long[ Count ] = Mid
Count += 1
UNDO; UNDO after finding a long.
- - .
return
This table lists the smallest Seed that produces a run the a given length. All lengths up through 471 are covered. Additional Seeds known to have longer runs are also listed. Seeds denoted with an asterisk have not been shown to be the lowest and are unlikely to be so.
Run Lowest Run Lowest Run Lowest
Length Seed Length Seed Length Seed
200 3718_71359 400 21799_62942_60379
1 1 201 2479_14239 401 21405_11748_07711
2 3 202 1652_76159 402 5202_05836_02919
3 11 203 21736_15775 403 9415_56398_74495
4 7 204 2938_24283 404 19377_44837_87003
5 39 205 1958_82855 405 12918_29891_91335
6 287 206 6207_52511 406 17224_39855_88447
7 231 207 3482_36187 407 12684_51406_26791
8 191 208 4138_35007 408 8456_34270_84527
9 127 209 16511_71495 409 3681_03871_92999
10 359 210 1274_56255 410 5637_56180_56351
11 511 211 2452_35559 411 89498_75294_55855
12 239 212 54256_72039 412 93998_85108_63087
13 159 213 2179_87163 413 5741_46618_62815
14 639 214 2906_49551 414 17459_12091_07615
15 283 215 1937_66367 415 5939_15976_23151
16 991 216 1453_24775 416 15519_21858_73435
17 251 217 968_83183 417 8064_83041_80415
18 167 218 15835_07967 418 6763_37123_94239
19 111 219 6613_98811 419 4508_91415_96159
20 1695 220 19681_65887 420 48746_49581_33967
21 1307 221 20794_41767 421 4007_92369_74363
22 871 222 3266_10023 422 24373_24790_66983
23 927 223 9840_82943 423 78236_45368_30463
24 671 224 12322_61787 424 39780_94748_90239
25 155 225 6560_55295 425 41352_12952_02335
26 103 226 28484_61311 426 27568_08634_68223
27 1639 227 4093_44047 427 5629_78582_46567
28 91 228 2728_96031 428 3753_19054_97711
29 3431 229 1819_30687 429 2502_12703_31807
30 3399 230 13046_21055 430 1668_08468_87871
31 2287 231 13249_21887 431 93722_74228_23271
32 71 232 955_92191 432 2965_48389_11771
33 6395 233 11041_80463 433 1976_98926_07847
34 47 234 53284_87839 434 1317_99284_05231
35 31 235 1_35512_07911 435 28943_78244_38015
36 2047 236 90341_38607 436 30596_65882_27631
37 27 237 637_28127 437 19295_85496_25343
38 1819 238 1_52182_80607 438 13598_51503_23391
39 17691 239 1_71086_56891 439 82737_64890_38527
40 6887 240 32463_39311 440 12087_56891_76347
41 4591 241 21642_26207 441 8058_37927_84231
42 13439 242 14428_17471 442 4_23915_18245_88635
43 6383 243 2_04459_54119 443 7163_00380_30427
44 4255 244 1_36306_36079 444 4775_33586_86951
45 7963 245 90870_90719 445 6367_11449_15935
46 7527 246 60580_60479 446 64112_39736_21607
47 12399 247 1_80196_82047 447 1_07262_39342_85695
48 7279 248 1_78250_84863 448 56988_79765_52539
49 1583 249 2177_40015 449 3183_55724_57967
50 1055 250 18014_87687 450 50656_70902_68923
51 703 251 12009_91791 451 33771_13935_12615
52 15039 252 1_66709_63135 452 47672_17485_71419
53 1_11259 253 62505_17663 453 1_66375_99560_77927
54 41407 254 3_20605_07419 454 2_21834_66081_03903
55 62079 255 1_48843_35615 455 7066_59241_17439
56 77031 256 8_71471_71839 456 1_47889_77387_35935
57 94959 257 62160_83103 457 4_11280_14211_95471
58 34239 258 87814_12679 458 1_10917_33040_51951
59 1_38751 259 2_40839_89231 459 73944_88693_67967
60 99007 260 2_39626_04007 460 8_11790_31161_74975
61 1_06239 261 2_14079_90427 461 2_07735_42996_78459
62 1_87327 262 58542_75119 462 6_50339_74304_30023
63 69375 263 39028_50079 463 8_83007_18992_87271
64 2_26767 264 34_94140_71423 464 5_88671_45995_24847
65 1_04303 265 2_52445_54015 465 7_84895_27993_66463
66 10087 266 8_17745_57807 466 3_92447_63996_83231
67 2_56511 267 6_01420_63643 467 7_13715_33954_73519
68 67583 268 41116_44527 468 8_41811_83220_39807
69 90111 269 54821_92703 469 11_22415_77627_19743
70 45055 270 27410_96351 470 4_64802_52950_28271
71 1_26575 271 2_37598_27611 471 5_63923_23124_72903
72 2_99259 272 13_98691_68255 472 3376_21358_59810_79551 *
73 96383 273 18273_97567 473 3_09868_35300_18847
74 3_36199 274 2_81597_95687 474 5_33204_30582_22495
75 64255 275 5_98341_74399 475 5_68559_52882_94911
76 84383 276 47047_65167 476 5_88976_68639_81599
77 57115 277 62730_20223 477 7_85302_24853_08799
78 56255 278 3_98894_49599 478 3_64834_92654_93159
79 37503 279 5_31859_32799 479 7_04108_99411_33671
80 60975 280 31365_10111 480 3_92651_12426_54399
81 45127 281 8_51539_52959 481 2_61767_41617_69599
82 3_93967 282 27880_08987 482 1_74511_61078_46399
83 4_23679 283 2_34400_29087 483 4117_16887_58947_34847 *
84 17_59951 284 33_86694_20543 484 ?
85 35655 285 3_73541_80511 485 ?
86 4_34223 286 4_98055_74015 486 ?
87 4_95687 287 2_49027_87007 487 7_77698_13233_89979
88 6_65215 288 5_22618_69567 488 6467_11939_14597_70367 *
89 16_43759 289 7_52598_71231 489 4_26419_64662_21183
90 5_28895 290 5_01732_47487 490 2_84279_76441_47455
91 7_30559 291 15_33354_91615 491 4_60858_15249_71839
92 4_37247 292 3_49800_46495 492 3_07238_76833_14559
93 21_62111 293 33_12246_89767 493 13_84890_41570_70619
94 4_32923 294 18_88908_83743 494 2_30429_07624_85919
95 5_65247 295 21_53848_33215 495 1_53619_38416_57279
96 2_88615 296 16_79030_07771 496 2_04825_84555_43039
97 3_76831 297 14_44607_75535 497 1_02412_92277_71519
98 25_48479 298 12_52916_45607 498 1_82067_41827_04923
99 6_11455 299 19_62812_97639 499 4198_10985_60470_38463 *
100 6_08111 300 106_36415_82407 500 2798_73990_40313_58975 *
101 15_85403 301 70_90943_88271 501 ?
102 4_05407 302 47_36445_47375 502 ?
103 2_70271 303 38_01037_73863 503 ?
104 3_62343 304 31_57630_31583 504 ?
105 4_01151 305 28_43969_52295 505 ?
106 15_63647 306 3_39805_39439 506 11_68098_49806_68831
107 10_42431 307 15_01644_53871 507 7_78732_33204_45887
108 67_21703 308 2_26536_92959 508 ?
109 3_81727 309 22_47087_03047 509 13_84413_03474_59355
110 6_67375 310 2_01366_15963 510 ?
111 6_26331 311 19_97410_69375 511 ?
112 16_91807 312 115_02840_49727 512 ?
113 15_64063 313 2_38656_18919 513 ?
114 15_41147 314 62_03986_72495 514 ?
115 10_27431 315 2_12138_83483 515 ?
116 11_27871 316 14_98058_02031 516 ?
117 19_91615 317 76_68560_33151 517 11_92179_57947_49439
118 13_27743 318 113_17793_53631 518 ?
119 73_03711 319 9_98705_34687 519 ?
120 62_55855 320 47_83372_65823 520 2566_05158_24781_23007 *
121 64_92187 321 56_69182_40975 521 9_41969_05044_93383
122 78_49755 322 42_51886_80731 522 6_27979_36696_62255
123 31_37471 323 14_02845_37063 523 6_07013_04245_19807
124 92_94427 324 18_89727_46991 524 ?
125 84_84287 325 25_19636_62655 525 ?
126 27_88863 326 16_79757_75103 526 9_92362_45644_04551
127 74_99935 327 365_42187_33311 527 0
128 60_79559 328 12_59818_31327 528 12_78973_98245_02639
129 62_04543 329 56_66198_06719 529 ?
130 208_08639 330 225_38353_49759 530 11_36865_76217_80123
131 99_41863 331 132_57301_44347 531 ?
132 292_56191 332 164_95313_56143 532 10_10547_34415_82331
133 88_37211 333 117_65490_20911 533 ?
134 20_91647 334 128_74025_86111 534 ?
135 13_94431 335 2_61309_34783 535 ?
136 173_92879 336 130_33334_17199 536 ?
137 130_02751 337 8_39878_87551 537 ?
138 74_60635 338 88_19891_93575 538 13_80641_46179_69947
139 25_33535 339 2_06466_64519 539 ?
140 16_89023 340 126_76301_41951 540 12_27236_85493_06619
141 11_26015 341 1_83525_90683 541 ?
142 649_93051 342 2_44701_20911 542 12_005_442_185_411_428_351 *
143 199_25503 343 88_38200_96231 543 ?
144 137_74695 344 2_17512_18587 544 ?
145 92_80639 345 1_22350_60455 545 8_18157_90328_71079
146 460_43247 346 508_63175_09375 546 ?
147 282_90175 347 89_86963_69947 547 7_27251_46958_85403
148 573_30463 348 1857_01714_67519 548 ?
149 548_70655 349 515_71428_56607 549 ?
150 463_55695 350 401_88187_72839 550 ?
151 487_73915 351 550_96077_10143 551 5_74618_44510_69947
152 325_15943 352 1068_14653_56287 552 3_83078_96340_46631
153 419_46879 353 712_09769_04191 553 5_10771_95120_62175
154 121_32095 354 474_73179_36127 554 ?
155 80_88063 355 724_40525_17375 555 2_55385_97560_31087
156 216_77295 356 1975_46755_54139 556 1_70257_31706_87391
157 143_78779 357 1316_97837_02759 557 ?
158 419_42559 358 1755_97116_03679 558 1_51339_83739_44347
159 2416_82847 359 1238_01143_11679 559 1_00893_22492_96231
160 408_14363 360 825_34095_41119
161 1873_75615 361 1802_69767_67615
162 1318_01135 362 552_13957_48159
163 441_86399 363 801_19896_74495
164 294_57599 364 414_10468_11119
165 392_76799 365 712_17685_99551
166 196_38399 366 3150_81354_71707
167 532_71551 367 276_06978_74079
168 710_28735 368 3350_85300_61951
169 272_09575 369 1830_70676_99951
170 355_14367 370 1220_47117_99967
171 601_12511 371 281_35382_12167
172 400_75007 372 2383_89422_84287
173 534_33343 373 250_09228_55259
174 1430_61311 374 2872_96231_36495
175 5007_52231 375 2268_60535_77471
176 1626_12223 376 2553_74427_87995
177 1072_95983 377 1702_49618_58663
178 226_49071 378 4299_29337_89863
179 715_30655 379 3075_01027_27359
180 201_32507 380 8939_48210_46783
181 134_21671 381 312_26276_52839
182 4013_06907 382 208_17517_68559
183 2792_00511 383 6732_18619_93087
184 206_38335 384 7055_82387_40647
185 2724_73947 385 17070_06081_85759
186 7579_16519 386 2467_79053_34271
187 8367_10559 387 12516_43323_86791
188 267_16671 388 2599_81224_92071
189 1440_91295 389 17844_30644_42727
190 1921_21727 390 14939_44647_67771
191 960_60863 391 10752_86948_05735
192 640_40575 392 34918_79117_77919
193 3402_08287 393 4722_37607_06159
194 569_24955 394 3148_25071_37439
195 5250_68415 395 2636_51570_99263
196 4315_57735 396 8778_47348_29927
197 14509_55711 397 1_02731_17849_34043
198 3836_06875 398 7803_08754_04379
199 6346_28095 399 10404_11672_05839