The lowest Seed for runs with up to 471 Odd steps were found.
This was sped up by filtering out Seeds producing only trivial runs.
The graph below puts the run length on the horizontal access and
Characterizing The Collatz Sequence
Bradley Berg May 29, 2026
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 / 2| + N + 1
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 to 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 through step i. 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 defined as the final value of Ki in a run.
Seed * 3^L + DL
NL = --------------- < Seed
2^KL
Since the values of Ni only drops below the Seed at the end of
a run then
3^L + DL / Seed
---------------- < 1
2^KL
Here the term
3^L < 2^KL and KL >= ⌈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
The value of
The length of a run, L, is the number of terms in the expansion. Since there are differnt interpretations of K for runs with the same number of terms, the number of Even transitions is not a consistent metric for the run length. For example, runs with Seeds of 608_111 and 3_124_719 both have 100 terms. However the value of K for 608_111 is 159 and for 3_124_719 it is 160.
Here is the algebraic expansion for a longer run with a Seed of 359.
The run length is 3 and K is 6,
but
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
Depending on the context there are three definitions for K values.
Ki Total number of even steps after each transition KL = K Total number of even steps in a run Ko Ko = ⌈log2( 3^L )⌉
In this section we derive these upper and lower bounds for 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 transition.
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 Ki. Should Ki get too big, the corresponding value in the run dips below the horizon and it 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.
To stay above the horizon
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 27_328 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. >
- -
Low Order Top Length Number Of Long Percentage Of Bit Width Eliminated Runs Remaining Runs Remaining 20 12 27_328 5.21% 21 13 46_611 4.45% 22 13 93_222 4.45% 23 14 168_807 4.02% 24 15 286_581 3.42% 25 15 573_162 3.42% 26 16 1_037_374 3.09% 27 17 1_762_293 2.63% 28 17 3_524_586 2.63% 29 18 6_385_637 2.38% 30 18 12_771_274 2.38% 31 19 23_642_078 2.20% 32 20 41_347_483 1.93% 40 25 6_402_835_000 1.16% 48 30 TBD 0.29% est
Long runs will initially have more Odd than Even transitions. Odd transitions move the sequence up and Even transitions move it down. Once a run gets beyond the initial phase its values become more randomized, balancing out Even and Odd transitions. The smallest Seed producing a given run length is the biggest outlier. As such they are needed to define outer bounds for the sequence.
The lowest Seed for runs with up to 471 Odd steps were found.
This was sped up by filtering out Seeds producing only trivial runs.
The graph below puts the run length on the horizontal access and
Trend line for the log2( Seed ) 0.0865 * L + 10.86 Trend line for the Seed 1858.6 * 1.061^L
Despite substantial variance the overall trend is linear and the Seed is exponential with respect to the length. The chart could give the impression that longer runs can continue as Seeds get larger. That's one way it can play out, but another outcome could be that some longer run lengths cease to exist and eventually limit is reached on run lengths.
With larger Seeds longer runs become more sparse. A smaller pecentage of runs continue to produce ever higher run lengths. We see that longer runs using larger Seeds can navigate more paths to avoid values that swiftly terminate. Wider Seed have more possible combinations and some result in longer runs. Even though there are more combinatins, there is a finite number of potential paths for a given Seed width. Some higher run lengths may never be reached by any of those paths. For a given Seed width eventually there is a maximum run length.
As runs get longer and the values get wider the total number of bits in play increases. More bits in a randomized process diminish irregularities as runs get longer. As a run progresses Even and Odd transitions balance out to push the sequence down and limit its length. Only at the start of a long run can Odds transitions can dominate over Evens. Once the information in the Seed is lost the run becomes fully randomized and Even and Odd transitions average out.
The next few groups of long runs are based on the three longest runs found
by
Seed/2 2*Seed+1 (Seed*3+1)/2 (Seed*2-1)/3
Length Seed Seed
605 811_58736_12332_33375 #b43_5667_b280_e5df
606 405_79368_06166_16687 #5a1_ab33_d940_72ef
607 541_05824_08221_55583 #782_399a_7700_993f
608 270_52912_04110_77791 #3c1_1ccd_3b80_4c9f
609 180_35274_69407_18527 #280_bdde_27aa_ddbf Roosendaal
615 8784_16162_88632_68635 #79e7_9c76_0203_071b
616 11712_21550_51510_24847 #a28a_25f2_ad59_5ecf
617 7808_14367_01006_83231 #6c5c_194c_7390_e9df
623 1854_70828_47202_68527 #19bd_3f5b_b8b9_c4ef
624 1236_47218_98135_12351 #1128_d4e7_d07b_d89f Silva
630 5856_10775_25755_12423 #5145_12f9_56ac_af67
632 3904_07183_50503_41615 #362e_0ca6_39c8_74ef
634 2602_71455_67002_27743 #241e_b319_7bda_f89f Silva
Seeds with only a few low order one bits cannot produce exceptionally
long runs no matter their size. Once the first Even transition in a run is
reached, subsequent values in the run soon become highly randomized.
In turn this drives the sequence below the Seed regardless of how large
the higher order bits get,
Seeds with a number of low order one bits begin a run with an Odd transition for each bit. After the low one bits are exhausted the remaining bits are scrambled and the value of the sequence is much higher than the original Seed. The run now has to fall further before it goes below the Seed. Such runs can get arbitrarilly long. We refer to Seeds with many low order one bits as Beans.
A Bean in binary is: [<High bits> | 0 | <Low one bits>]
Parameters are {Shi, Slo} where:
Shi, the high order bits, is an even number including the zero bit delimiter.
Slo is the number of low order one bits.
Initially they transition from the Seed up to: (Shi + 1) * 3^Slo - 1
These runs act like a roller coaster that use a chain to drag carts up high and then take a bumpy ride to the bottom. They belong to a secondary class of runs and can potentially reach any desired length. Very long runs start with a sufficient number of low order ones along with select high order bits. Here are some long runs using Seeds that only contain one bits.
Length = 2625 Seed = 2^997 - 1 Length = 24632 Seed = 2^9835 - 1 Length = 242923 Seed = 2^99797 - 1
Run lengths for Beans with high order bits up to 32 bit wide were computed.
Their graph shows the log2 of Beans with various numbers of low order
ones that produced the longest lengths
Trend line for the log2( Bean ) 0.324 * L - 95.26 Trend line for the Bean 2.108e-29 * 1.2518^L
This next graph compares Seeds and Beans over run lengths up to 470 steps. It show how Beans diverge from Seeds and get substantially larger for the same run length. In a few of the smallest cases they are the same. While lengths for Seeds eventually end, Beans can extend out to any length.
Once the initial Odd transitions are applied the run reaches a peak value.
From there it declines until it goes down below the Seed.
The average gain of the remaining transitions is computed by splitting the
total gain into a product of average gains over the length.
This chart shows the average gain for run lengths of Beans from
The smallest Bean producing a given run length will have the largest average gain. That is, a run with a larger gain will decline more slowly and wind up with more steps. However, from the chart we see that the average gain drops as Shi increases. Entropy increases as more bits are in play due to the width of the Seed and the length of the run. As runs get longer entropy appoaches one; pushing down the average gain. In the limit runs become so regular that the only remaining determinant of length is Slo.
To illustrate the progression of runs in a Bean,
This shows that like smaller Seeds, Beans with the same value of Shi do not grow monotonically. Eventually they should top out as well as they are subject to the same forces. Since Beans are wider and the run lengths are longer they will tend to have even more regular behaviour.
Seed Maximum Width Length Seed in Decmal Seed in Hexadecimal 76 515 448_07594_01188_30283_48927 #97d_064828_ffffffffff 77 529 1014_64900_87639_87432_57087 #157c_6c5d42_ffffffffff 78 499 2671_44962_35917_71921_44895 #3891_f519fe_ffffffffff 79 569 4115_17766_77028_30273_65887 #58df_a058ca_ffffffffff 80 623 10497_23535_45096_57967_16543 #de49_a07b5a_ffffffffff 81 553 20204_84938_53655_79350_67135 #1_abda_b53ef2_ffffffffff 82 580 30786_98240_33354_88966_49215 #2_8bf0_90e754_ffffffffff 83 630 ? 87680_73036_51559_01639_22943 #7_40b6_31158c_ffffffffff 84 597 ? 97055_37592_27379_59204_29055 #8_073a_3f442e_ffffffffff
The length of a run, L, is the number of Odd transitions until the sequence goes below the Seed. The algebraic expansion of a full run is:
Seed * 3^L + DL L
NL = --------------- where: DL = ∑ 3^(L-j) * 2^Kj-1
2^EL j=1
Together the lower and upper bounds on DL are:
3^L - 2^L <= DL <= L * 3^(L-1)
Runs with trivialy short lengths are easily dismissed by checking the low order bits.
For the first several hundred run lengths the trend line for the lowest Seeds are exponential with respect to the run length.
log2( Seed ) = 0.0865 * L + 10.86
A Bean in binary is: [<High bits> | 0 | <Low one bits>]
Parameters are {Shi, Slo} where:
Shi, the high order bits, is an even number including the zero bit delimiter.
Slo is the number of low order one bits.
Only Seeds with many low order one bits produce long runs. Trend lines for Beans are also exponential with the run length. The trend line for the log2( Bean ) is:
log2( Bean ) = 0.324 * L - 95.26
The primary finding is the importance of Beans in understanding the Collatz sequence to the exclusion of other Seeds.
[1] John H. Conway, "Unpredictable iterations". Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49-52.
[2] Tomas Oliveira e Silva,
"Maximum Excursion and Stopping Time Record-Holders for the 3x+1 Problem: Computational Results,"
Mathematics of Computation, vol. 68, no. 225, pp. 371-384, Jan. 1999.
[3] Eric Roosendaal, "3x+1 Glide Records"
[4] Bradley A. Berg, "Entropy Makes the Collatz Sequence Go Down".
https://www.researchgate.net/publication/382214052_Entropy_Makes_the_Collatz_Sequence_Go_Down
[5] Techneon, "Gilda Language Reference", https://gilda.techneon.com/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
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:
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. They are denoted with an asterisk and are Beans that are unlikely to be the smallest Seed.
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 17_89198_63775_55583
84 17_59951 284 33_86694_20543 484 18857_81175_66231_59025_99167 *
85 35655 285 3_73541_80511 485 12571_87450_44154_39350_66111 *
86 4_34223 286 4_98055_74015 486 2_74133_05463_23521_06267 *
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 8_45920_87172_49292_37605_41695 *
102 4_05407 302 47_36445_47375 502 75_36021_25074_41496_06399 *
103 2_70271 303 38_01037_73863 503 2_85208_53105_79003_28163_73759 *
104 3_62343 304 31_57630_31583 504 2622_50471_60697_99345_52063 *
105 4_01151 305 28_43969_52295 505 1_06953_19914_67126_23061_40159 *
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 508 422_08205_85757_66521_38128_34303 *
109 3_81727 309 22_47087_03047 509 13_84413_03474_59355
110 6_67375 310 2_01366_15963 510 92879_13404_58959_90968_31999 *
111 6_26331 311 19_97410_69375 511 61919_42269_72639_93978_87999 *
112 16_91807 312 115_02840_49727 512 941_73687_22009_74130_56844_26751 *
113 15_64063 313 2_38656_18919 513 3144_12632_21095_19560_65282_94911 *
114 15_41147 314 62_03986_72495 514 16_10919_38167_23951
115 10_27431 315 2_12138_83483 515 5961_04620_52329_16316_30868_80767 *
116 11_27871 316 14_98058_02031 516 17_88269_36921_24159
117 19_91615 317 76_68560_33151 517 11_92179_57947_49439
118 13_27743 318 113_17793_53631 518 139_19500_73564_51050_10612_63359 *
119 73_03711 319 9_98705_34687 519 18_972_360_957_990_559
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 2484_25636_96633_90157_19769_86623 *
125 84_84287 325 25_19636_62655 525 6624_68365_24357_07085_86052_97663 *
126 27_88863 326 16_79757_75103 526 9_92362_45644_04551
127 74_99935 327 365_42187_33311 527 466_52869_55572_50967_87137_00351 *
128 60_79559 328 12_59818_31327 528 12_78973_98245_02639
129 62_04543 329 56_66198_06719 529 55476_15515_11317_95272_25676_92287 *
130 208_08639 330 225_38353_49759 530 11_36865_76217_80123
131 99_41863 331 132_57301_44347 531 33744_71159_02077_28248_90768_09727 *
132 292_56191 332 164_95313_56143 532 10_10547_34415_82331
133 88_37211 333 117_65490_20911 533 362_42376_76532_17405_48466_27839 *
134 20_91647 334 128_74025_86111 534 1_59953_94239_28968_58844_58447_66719 *
135 13_94431 335 2_61309_34783 535 735_59028_54093_04321_90738_59583 *
136 173_92879 336 130_33334_17199 536 15_53221_64452_16191
137 130_02751 337 8_39878_87551 537 13_86839_45011_56591_46375_87521_20831 *
138 74_60635 338 88_19891_93575 538 13_80641_46179_69947
139 25_33535 339 2_06466_64519 539 32_63046_12749_63915_46363_41546_84415 *
140 16_89023 340 126_76301_41951 540 12_27236_85493_06619
141 11_26015 341 1_83525_90683 541 8236_54981_57304_17613_97579_44831 *
142 649_93051 342 2_44701_20911 542 12005_44218_54114_28351 *
143 199_25503 343 88_38200_96231 543 1630_19949_77081_92117_09340_71295 *
144 137_74695 344 2_17512_18587 544 740_13630_83975_26933_29145_36447 *
145 92_80639 345 1_22350_60455 545 8_18157_90328_71079
146 460_43247 346 508_63175_09375 546 1_90557_91362_45110_63975_33639_80287 *
147 282_90175 347 89_86963_69947 547 7_27251_46958_85403
148 573_30463 348 1857_01714_67519 548 80569_99868_92139_41538_21338_33727 *
149 548_70655 349 515_71428_56607 549 6_46445_75074_53691
150 463_55695 350 401_88187_72839 550 53713_33245_94759_61025_47558_89151 *
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 554 914_63196_27133_87077_46815 *
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 52_20413_95430_21187_36054_27648_26623 *
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 560 2_14763_29372_98076_80769_09311_42655 *
161 1873_75615 361 1802_69767_67615 561 472_91610_64566_44447_12232_00451_13343 *
162 1318_01135 362 552_13957_48159 562 123_25737_59998_68405_93411_03153_80735 *
163 441_86399 363 801_19896_74495 563 164_34316_79998_24541_24548_04205_07647 *
164 294_57599 364 414_10468_11119 564 95278_95681_22555_31987_66819_90143 *
165 392_76799 365 712_17685_99551 565 321_63639_22219_51853_30995_82935_53151 *
166 196_38399 366 3150_81354_71707 566 1_01655_04111_04009_73077_63964_64127 *
167 532_71551 367 276_06978_74079 567 67770_02740_69339_82051_75976_42751 *
168 710_28735 368 3350_85300_61951 568 6_54636_98218_28496_04626_10901_23775 *
169 272_09575 369 1830_70676_99951 569 118_30368_88517_91519 *
170 355_14367 370 1220_47117_99967 570 1979_25961_53177_42308_22270_54998_97855 *
171 601_12511 371 281_35382_12167 571 16_00456_94467_30190_25203_75018_78271 *
172 400_75007 372 2383_89422_84287 572 64_43885_95661_24027_24301_42471_53663 *
173 534_33343 373 250_09228_55259 573 1657_87495_22034_84757_15405_06637_76255 *
174 1430_61311 374 2872_96231_36495 574 1054_51688_94828_84539_37935_15519 *
175 5007_52231 375 2268_60535_77471 575 256_48223_52190_94117_03058_95067_81183 *
176 1626_12223 376 2553_74427_87995 576 527_25844_47414_42269_68967_57759 *
177 1072_95983 377 1702_49618_58663 577 703_01125_96552_56359_58623_43679 *
178 226_49071 378 4299_29337_89863 578 75_75038_47471_19702_62267_91797_22751 *
179 715_30655 379 3075_01027_27359 579 3901_41912_66397_90309_41740_05990_52287 *
180 201_32507 380 8939_48210_46783 580 975_35478_16599_47577_35435_01497_63071 *
181 134_21671 381 312_26276_52839 581 128_24111_76095_47058_51529_47533_90591 *
182 4013_06907 382 208_17517_68559 582 251_10039_78126_78098_45423_32022_94783 *
183 2792_00511 383 6732_18619_93087 583 282_9669_54304_36592_87261_10373_15071 *
184 206_38335 384 7055_82387_40647 584 351_50562_98276_28179_79311_71839 *
185 2724_73947 385 17070_06081_85759 585 955_95310_4663_47279_09239_91391_92831 *
186 7579_16519 386 2467_79053_34271 586 64_12055_88047_73529_25764_73766_95295 *
187 8367_10559 387 12516_43323_86791 587 5381_48226_93449_34033_85336_98633_72799 *
188 267_16671 388 2599_81224_92071 588 1705_91772_10944_83642_44444_53092_06527 *
189 1440_91295 389 17844_30644_42727 589 263_62922_23707_21134_84483_78879 *
190 1921_21727 390 14939_44647_67771 590 2412_52165_63535_80924_85110_06909_39903 *
191 960_60863 391 10752_86948_05735 591 546_19237_41546_06637_81706_64767_97951 *
192 640_40575 392 34918_79117_77919 592 18_42468_24855_97100_38845_33364_81791 *
193 3402_08287 393 4722_37607_06159 593 175_75281_49138_14089_89655_85919 *
194 569_24955 394 3148_25071_37439 594 1_71180_47741_82753_79730_48319_67231 *
195 5250_68415 395 2636_51570_99263 595 1_20153_03646_59369_52611_38846_24167_89503 *
196 4315_57735 396 8778_47348_29927 596 131_81461_11853_60567_42241_89439 *
197 14509_55711 397 1_02731_17849_34043 597 87_87640_74569_07044_94827_92959 *
198 3836_06875 398 7803_08754_04379 598 2924_07691_65693_45972_15208_48411_03359 *
199 6346_28095 399 10404_11672_05839 599 42795_11935_45688_44932_62079_91807 *
Runs were computed for Beans with various values of Slo. The longest run is listed for each of them. The average gain for each Odd transition is then computed. The log2 of the Bean and DL for the run is also listed.
Bean = Shi * 2^Slo + 2^Slo - 1 Slo is the number of low order one bits.
Peak = (Shi + 1) * 3^Slo - 1 Shi has the upper 32 bits and is even.
NL = Peak * GT The last value in a run is the Peak times the total gain.
GT = NL / Peak Total gain
GT = GAvg^Length The total gain is the product of all average gains.
GAvg = (NL / Peak)^(1 / Length) Average gain per odd step
Slo Shi Length GAvg Log2(Bean) Log2(DL)
25 3120_58388 412 0.972 53.21724 654.22569
26 16598_56966 417 0.973 56.62841 661.77633
27 16598_56966 417 0.972 57.62841 662.30645
28 19590_98510 450 0.974 58.86754 713.91923
29 6530_32836 451 0.974 58.28258 715.30964
30 6530_32836 451 0.973 59.28258 715.67560
31 30114_87140 488 0.974 62.48783 776.36217
32 10038_29046 490 0.973 61.90287 779.12575
33 10038_29046 482 0.974 62.90287 766.96420
34 36461_67496 458 0.974 65.76373 727.12904
35 21320_65256 472 0.965 65.98960 749.96838
36 7106_88418 473 0.965 65.40464 751.15382
37 7106_88418 473 0.966 66.40464 751.86589
38 31621_71770 467 0.965 69.55827 741.56993
39 25560_53068 470 0.965 70.25127 746.14789
40 3513_52418 471 0.961 68.38834 747.62769
41 1171_17472 472 0.961 67.80338 748.92821
42 25702_39274 499 0.965 73.25926 792.82726
43 8567_46424 502 0.964 74.25926 754.65341
44 28655_25598 479 0.959 75.41615 759.89272
45 39288_80632 630 0.966 76.87147 998.92642
46 39288_80632 630 0.968 77.87147 999.23928
47 6498_85096 554 0.963 76.27561 879.28772
48 6498_85096 554 0.964 77.27561 879.93873
49 34448_74948 497 0.958 80.68180 790.55660
50 10901_99742 493 0.954 80.02195 781.99490
51 41246_62124 510 0.952 82.94163 808.35997
52 13748_87374 511 0.952 82.35667 809.93530
53 13748_87374 511 0.951 83.35667 809.95450
54 7916_12694 504 0.954 83.56022 799.18389
55 7916_12694 503 0.953 84.56022 797.24449
56 32919_79986 538 0.949 87.61631 853.79092
57 32919_79986 538 0.950 88.61631 854.40193
58 11758_51558 545 0.954 88.13106 865.82886
59 22866_19004 596 0.956 90.09057 946.48124
60 7622_06334 597 0.956 89.50561 947.66940
61 7622_06334 593 0.957 90.50561 941.67956
62 34807_88214 584 0.954 91.50561 927.07037
63 7622_06334 577 0.953 92.50561 915.32773
64 5891_55278 547 0.949 93.13407 867.85012
65 19790_81428 558 0.948 95.88218 885.51494
66 19790_81428 555 0.949 96.88218 880.60992
67 31055_55874 552 0.951 98.53220 876.20028
68 14499_55042 599 0.950 98.43336 950.00152
69 14499_55042 599 0.945 100.76059 889.44842
70 14499_55042 594 0.948 100.43336 941.73341
71 14499_55042 594 0.952 101.43336 941.95769
72 25303_86798 600 0.945 103.23671 951.83733
73 20410_17300 617 0.948 103.92664 979.88793
74 20727_22558 614 0.945 104.94888 974.19627
75 38812_41816 627 0.947 106.85387 995.08337
76 38381_41610 627 0.947 107.83776 994.40264
77 12793_80536 629 0.946 107.25280 997.64295
78 42424_40766 648 0.946 109.98225 1027.91802
79 1421_53392 632 0.945 106.08287 1002.21495
80 31339_41778 658 0.945 111.54533 1044.18879
81 31339_41778 658 0.946 112.54533 1044.85708
82 31339_41778 654 0.944 113.54533 1038.90226
83 41086_99820 632 0.941 114.93603 1003.07075
84 37016_03918 650 0.941 115.78550 1031.02783
85 12338_67972 651 0.941 115.20054 1032.39155
90 38130_67078 680 0.941 121.82830 1079.53391
95 3070_12984 666 0.935 123.19372 1057.26532
100 1147_28848 680 0.937 126.77365 1079.32028
105 549_60244 698 0.930 130.71189 1106.48669
110 20327_08180 706 0.928 140.92076 1119.94641
115 31165_31866 732 0.927 146.53729 1160.83635
120 29413_13422 811 0.929 151.45381 1286.83124
125 41453_84592 803 0.929 156.94886 1274.15537
130 24954_09214 806 0.925 161.21663 1278.30610
135 40019_18080 817 0.924 166.89804 1295.81027
140 11344_76812 837 0.923 170.07938 1328.23030
145 2810_69890 839 0.919 173.06635 1330.87356
150 23229_15894 873 0.920 181.11329 1385.83694
155 10082_40524 888 0.918 184.90919 1408.70916
160 35827_19298 890 0.909 191.73841 1411.54305
165 29365_78612 900 0.913 196.45149 1426.64373
170 33889_04628 915 0.912 201.65817 1450.73500
175 38431_59930 932 0.912 206.83965 1478.08729
180 18278_78606 945 0.910 210.76752 1498.88718
185 29878_78330 1000 0.912 216.47647 1586.61117
190 10501_72108 959 0.905 219.96798 1521.05522
195 40037_62374 1020 0.909 226.89871 1617.04963
200 32546_48254 1076 0.912 231.59985 1706.02879
205 42614_42586 1082 0.913 236.98869 1716.13094
210 26018_69030 1111 0.910 241.27690 1762.73786
215 8672_89676 1075 0.903 244.69194 1704.66470
220 36319_07086 1131 0.906 251.75808 1792.76800
225 35077_58744 1087 0.900 256.70790 1723.65686
230 16421_38206 1102 0.899 260.61293 1747.65954
235 42318_56476 1145 0.901 266.97864 1815.54203
240 8216_70158 1170 0.901 269.61398 1854.59574
245 18275_89054 1164 0.898 275.76729 1845.90910
250 11787_15120 1170 0.897 280.13457 1854.63200
255 9455_05200 1184 0.894 284.81651 1877.02086
260 35146_06586 1247 0.899 291.71072 1978.17501
265 7914_72124 1235 0.895 294.55996 1958.05642
270 17606_85796 1258 0.896 300.71349 1994.39148
275 15567_03068 1230 0.890 305.53585 1950.70879
280 28726_42164 1290 0.895 311.41973 2045.50884
285 4894_92450 1319 0.895 313.86671 2091.24035
290 8253_42626 1366 0.896 319.62042 2165.12625
295 24760_27880 1359 0.894 326.20538 2155.04894
300 31054_89314 1376 0.893 331.53217 2181.68690
350 35131_19668 1486 0.883 381.71011 2356.67169
400 34589_93100 1624 0.877 431.68770 2575.12027
450 3153_86494 1826 0.875 478.23255 2894.33224
500 4903_56754 1917 0.867 528.86926 3038.69940
550 16574_51258 2163 0.871 580.62632 3429.38977
600 4993_34748 2223 0.861 628.89543 3524.40758
650 29322_75230 2386 0.857 681.44937 3782.47965
700 40758_52394 2569 0.859 731.92445 4072.70430
750 16193_35430 2656 0.852 780.59275 4210.93966
800 40726_15200 2798 0.851 831.92331 4435.52770
850 25043_61670 2883 0.844 881.22180 4570.26003
900 34900_12478 3135 0.850 931.70059 4969.43878
950 31381_68378 3255 0.846 981.54728 5160.26377
1000 31125_70566 3444 0.846 1031.53546 5458.72112
Beans with 40 low order one bits were computed up through 83 bits. This effort is currently still in progress.
Length Bean in Decimal Bean in Hexadecimal 500 6451_29778_82689_27768_33023 #889c_8e2dd6_ffffffffff 501 12013_03758_85112_32045_87519 #fe62_cea876_ffffffffff 502 1351_55626_43452_32285_36831 #1c9e_cd1acc_ffffffffff 503 3703_75533_22339_39994_54207 #4e6e_190f2c_ffffffffff 504 6487_08488_83988_70295_34719 #895e_8ec1da_ffffffffff 505 7860_15382_97853_86677_69855 #a671_fab2da_ffffffffff 506 1080_05873_57519_87745_25951 #16df_029870_ffffffffff 507 3160_91616_26625_44902_71743 #42ef_5c60dc_ffffffffff 508 1053_63872_08875_14967_57247 #164f_c9759e_ffffffffff 509 5878_63719_36940_34851_26655 #7c7c_27f512_ffffffffff 510 11188_90778_99550_92017_31583 #ecef_30cf9e_ffffffffff 511 4196_93428_62901_08882_61631 #58df_a058ca_ffffffffff 512 1398_97809_54300_36294_20543 #1d9f_e01d98_ffffffffff 513 19875_63965_79242_48710_67647 #1_a4e2_0ef8c2_ffffffffff 514 7743_66934_08120_24479_74399 #a3fa_83ff1e_ffffffffff 515 448_07594_01188_30283_48927 #97d_064828_ffffffffff 516 5403_13691_42564_83288_35071 #726a_76c25e_ffffffffff 517 8593_04635_26122_20249_04703 #b5f6_ff63f2_ffffffffff 518 52417_82449_73356_73070_55103 #4_55fd_989276_ffffffffff 519 30241_56461_64555_72674_31423 #2_8063_d9ac1e_ffffffffff 520 10976_33397_39854_97165_33247 #e86e_d3613e_ffffffffff 521 5527_27566_59610_40666_95167 #750b_6be27e_ffffffffff 522 18379_53157_91553_36984_00255 #1_8533_a3ed1e_ffffffffff 523 22752_28553_19682_79203_02079 #1_e1cc_631356_ffffffffff 524 10075_12700_44866_92262_37951 #d559_5f7872_ffffffffff 525 3321_13074_16545_40181_83167 #4653_e2c57e_ffffffffff 526 3043_94702_62919_62297_71263 #4075_4517c8_ffffffffff 527 8693_38542_18801_40150_86591 #b816_efd256_ffffffffff 528 27330_75201_88677_71282_55487 #2_42c0_4db6ee_ffffffffff 529 1014_64900_87639_87432_57087 #157c_6c5d42_ffffffffff 530 21360_22556_16001_42291_76319 #1_c452_03c97e_ffffffffff 531 1_00724_52286_16204_68121_59999 #8_54ec_bb881c_ffffffffff 532 22732_23210_11957_34067_44575 #1_e15f_ad598e_ffffffffff 533 52466_86265_16350_42678_21055 #4_5707_6eae9e_ffffffffff 534 28329_16464_44167_55859_45599 #2_57e4_b5921e_ffffffffff 535 20698_67796_55040_64341_93407 #1_b64f_c2121c_ffffffffff 536 14324_45935_02235_48883_92703 #1_2f55_0cbb7e_ffffffffff 537 12950_91441_15836_60831_53919 #1_123f_0c364c_ffffffffff 538 4316_97147_05278_86943_84639 #5b6a_59676e_ffffffffff 539 12976_94314_87647_05335_54175 #1_12cc_26575c_ffffffffff 540 5792_65069_75357_26163_06687 #7aaa_05b8f6_ffffffffff 541 86089_31062_99353_14852_90495 #7_1f03_168b58_ffffffffff 542 9086_02573_93046_14305_46431 #c067_71b8fe_ffffffffff 543 10922_41495_14728_80607_55967 #e74a_87b45c_ffffffffff 544 545 25473_44997_65060_76637_42975 #2_1b6b_d959c6_ffffffffff 546 28696_43687_66451_04950_96831 #2_5fab_b22e72_ffffffffff 547 8494_66786_31657_37495_42911 #b3e1_afcf6e_ffffffffff 548 4325_64771_62549_01778_51391 #5b99_621d1e_ffffffffff 549 1_05275_24883_43082_57748_09087 #8_b54a_447946_ffffffffff 550 48356_37331_49820_66708_80767 #3_fffc_6ccd54_ffffffffff 551 60614_54815_60967_38052_01407 #5_0390_1fbcd8_ffffffffff 552 99827_25807_65379_22917_49887 #8_41ec_a68090_ffffffffff 553 20204_84938_53655_79350_67135 #1_abda_b53ef2_ffffffffff 554 87922_11661_86569_27532_97407 #7_45d2_bfce04_ffffffffff 555 78309_35498_09065_77313_13663 #6_7a43_dd2dde_ffffffffff 556 557 558 559 93972_63945_90130_69467_81183 #7_c5f2_b303e6_ffffffffff 560 75558_37333_44355_12713_21599 #6_4002_c33950_ffffffffff 561 43032_99582_91956_91335_35231 #3_8f42_567c36_ffffffffff 564 33083_67151_25607_92890_24511 #2_bc92_f150c2_ffffffffff 565 25186_12444_48118_37571_07199 #2_1556_41131a_ffffffffff 569 4115_17766_77028_30273_65887 #58df_a058ca_ffffffffff 570 10728_20868_17544_46601_91231 #e32d_bc5d44_ffffffffff 572 77357_31799_73242_57648_96767 #6_661a_dc7470_ffffffffff 576 92360_94721_00064_66899_47647 #7_a3d1_b2b5fe_ffffffffff 577 87776_08184_38076_72420_27007 #7_42bb_17d7cc_ffffffffff 580 30786_98240_33354_88966_49215 #2_8bf0_90e754_ffffffffff 581 88959_89174_75280_45934_67391 #7_5bcc_8a1b9a_ffffffffff 587 45666_72615_82570_75781_63199 #3_c707_d18cd8_ffffffffff 591 89589_80409_27052_20266_55743 #7_6923_4d181e_ffffffffff 597 97055_37592_27379_59204_29055 #8_073a_3f442e_ffffffffff 623 10497_23535_45096_57967_16543 #de49_a07b5a_ffffffffff 630 87680_73036_51559_01639_22943 #7_40b6_31158c_ffffffffff