Entropy Makes the Collatz Sequence Go Down
Bradley Berg November 21, 2018
Edited June 18, 2024
ABSTRACT
We look at the Collatz Sequence from an information theory
perspective to lay out its underlying computational mechanics.
The mechanisms are similar to those used in pseudo random number
generators and one way hashes.
An individual run is divided into three phases. In the first
phase the influence of the seed runs its course after information
contained in the initial seed is lost. Values are randomized in
the second phase. They follow the statistical model where the
average gain is just over 0.866; causing them to decline. The
third phase begins once a value goes below the seed; providing
that the series is not circular. At this point we know the run
will terminate at one.
An equivalent restatement of the Collatz sequence steps through
alternating chains of even and odd values. This variation
constitutes a pseudo random number generator. The operations
used to scramble values are unbiased which results in an even
distibution of ones and zeros. The entropy of this mechanism is
high so that in the second phase values are fairly randomized.
Notations:
* multiplication operator
^ exponentiation operator
⊻ logical difference (exclusive or) operator
-> "Transition to" the the next value in a sequence.
The computational mechanics of the Collatz sequence are analyzed to determine the odds of taking an even or odd step. With the following definition of the sequence we show the the average odds of taking either step are even. In this case statistically the sequence will on average decline and eventually terminate. This variation is often referred to as the Syracuse sequence.
N is Even: N' = N / 2 N is Odd: N' = (3*N + 1) / 2
The gain of a each transition is its Output divided by the Input (N' / N). As N gets larger in the odd transition the "+ 1" term quickly becomes insignificant. To compute the gain for the odd transition in the limit we can safely drop the "+1"term.
Output / Input Gain
N is Even: {N / 2} / N 0.5
N is Odd: {(3*N + 1) / 2} / N 1.5
The total gain of a series of transitions is the product of the gains for each transition. Based on this the average gain of a sequence depends on the probability of taking either an odd or even path.
Gs = 1.5 ^ p(odd) * .5 ^ p(even)
The choice of which path is taken is determined by the low order bit of the input value. If the sequence should produces uniformly randomized values then the chances of taking either transitions is 50:50. This implies the low bit would need to be uniformally random over the sequence. The average gain of a uniformly randomized sequence is then:
Average transaction gain = <transaction gain> ^ p(<transaction>)
Average odd gain = 1.5 ^ .5 = 1.22474
Average even gain = 0.5 ^ .5 = 0.70711
Average sequence gain: Ga = 1.22474 * 0.70711 = 0.86603+
The statistical average gain in each step is less than one so on average the sequence declines. However, the gain in a single run for a given Seed can vary significantly. There could potentially still be a series where the the total gain indefinitely exceeds one and never terminates.
Breakeven Gain = 1.5 ^ p * 0.5 ^ (1 - p) = 1
ln( 1.5 ^ p * 0.5 ^ [1 - p]) = ln( 1 ) = 0
= ln( 1.5 ^ p ) + ln( 0.5 ^ (1 - p)) = 0
= p * ln( 1.5 ) + (1 - p) * ln( 0.5 ) = 0
p * 0.40546 = -(1 - p) * -0.69315
0.40546 / -0.69315 = -(1 - p) / p
-0.58497 = -1 / p + 1
-0.58497 - 1 = -1 / p
p = 1 / 1.58497 = 0.63093+
Gain = 1.5^0.63 * 0.5^0.37 = 0.99898 Just under breaking even
Gain = 1.5^0.64 * 0.5^0.36 = 1.01001 Just over breaking even
For any series of values to continually increase and never terminate it would have to sustain an average gain over one. To break even odd transitions would need to occur about 64% of the time. They would need to be applied over 1.7 times more than evens; which is substantially skewed. It remains to be shown that the sequence does not intrinsically favor odd transitions.
Consecutive iterations of the same kind of transition in a run form a chain. Even chains start with an even seed value that in binary have one or more trailing zeroes. After applying the transitions in an even chain the result simply has the low order zeros removed.
Odd chains consume an odd input and have multiple odd intermediate values. Eventually an Odd chain transitions to an even number. The number of consecutive low order one bits determines the chain length. For example, an input of 19 is a binary 10011 so the subsequent chain has two Odd transitions: 19 -> 29 -> 44
Let k be the number of low order one bits and
j is the input value with the low one bits removed plus 1.
The input to an odd chain has the form: j * 2^k - 1
The output of the chain simplifies to: j * 3^k - 1
N1 = (3*N + 1) / 2 First transition
= (3 * [j * 2^k - 1] + 1) / 2 Substitute N = j * 2^k - 1
= ([3 * j * 2^k - 3] + 1) / 2
= [3 * j * 2^k - 2] / 2
= 3^1 * j * 2^[k-1] - 1
Ni+1 = (3 * {3^i * j * 2^[k-i] - 1} + 1) / 2 Subsequent transitions
= ({9 * j * 2^[k-i] - 3} + 1) / 2
= {9 * j * 2^[k-i] - 2} / 2
= 3^[i+1] * j * 2^[k-i-1] - 1
Nk = 3^k * j - 1 Odd chain output
Each run of the Collatz sequence will have segments with alternating even and odd chains. For reference, here are the first few chains for the series beginning with a seed of 27.
Syracuse Sequence Even Odd j k base 2
27 -> 41 -> 124 -> 62 27 -> 62 7 2 110_11
-> 31 -> 31 31 1 11111_0
-> 47 -> 71 -> 107 -> 161 -> 242 -> 242 1 5 0_11111
-> 121 -> 121 121 1 111100_1
-> 364 -> 182 -> 182 61 1 1011011_0
-> 91 -> 91 91 1 10110_11
-> 137 -> 206 -> 206 23 2 1100111_0
-> 103 -> 103 103 1 1100_111
-> 155 -> 233 -> 350 -> 350 13 3 10101111_0
-> 175 -> 175 175 1 1010_1111
-> 263 -> 395 -> 593 -> 890 -> 890 11 4 110111101_0
Each run alternates between even and odd chains. We can represent this aspect algebraically by merging both into a single step. This gives us a series defined as a single transition. Since all intermediate values using this combined definition are even, an initial odd value needs to first transition one step using the "3*n + 1" rule to reach the first even number.
Every input has the binary form: j ones zeros
N = ([j + 1] * 2^ko - 1) * 2^kz -> [j + 1] * 3^ko - 1
where: kz - number of trailing zeros in N
ko - number of the next higher set of ones in N
j - N shifted right by kz + ko bits: N / 2^[kz + ko]
Using the previous example, initially we transition 27 to 82. From there the next few steps are:
j ko kz binary
82 -> 41 -> 124 -> 62 20 1 2 10100_10
-> 31 -> 484 -> 242 0 5 1 0_111110
-> 121 -> 364 -> 182 60 1 1 111100_10
-> 91 -> 274 -> 206 22 2 1 10110_110
-> 103 -> 310 -> 466 12 3 1 1100_1110
-> 233 -> 700 -> 350 116 1 1 1110100_10
-> 593 -> 1780 -> 890 10 4 1 1010_11110
Statistical averages only hold when the odds are fair. In this section we show why the dice are not loaded. Shannon entropy is a measure of information denoting the level of uncertainty about the possible outcomes of a random variable[1].
H = -p0 * log2( p0 ) + -p1 * log2( p1 )
where: p0 is the probability a bit is zero.
p1 is the probability a bit is one (1 - p0).
A set of coin tosses has p0 = p1 =.5 so its entropy is 1; totally random. When looking at the entropy of bits in a number then p0 is the percentage of zero bits. For the binary number, 1010_1111, p0 is .25 (H = 0.811). Strings of all ones or zeros have no entropy (H = 0). For a binary number we are measuring the bits in a number horizontally.
Bits in a series of numbers have two dimensions - horizontal bits in each individual value and vertical bits over the duration of the series. We can also measure entropy vertically over a number series. That means we can observe a select bit position in each value as the series progresses.
For Collatz the low order bit is of interest because it determines if a number is odd or even. In turn that determines which transition to take. When the entropy of the low order bit is high then on average there are nearly as many even transitions as odds.
Each kind of chain takes a value where the low order bits are a string of zeros or ones and either removes or replaces them. Even inputs remove low order zeros. The expression for odd chain inputs has a 2^k term that transitions to a 3^k term.
Since strings of zeros have no entropy and the j term has positive entropy, entropy increases each time an even transition is applied. In odd transitions entropy is also increased by removing the repeated ones and again by scrambling the remaining bits. The upper bits, j, are scrambled by multiplying j by a power of 3. As a run progresses this increase in entropy randomizes the values. The number of odd and even transitions balance out driving the sequence downward and eventually forcing it to terminate.
A Seed can be contrived to produce a run of any desired length. The longer the run the larger the seed has to be in order to contain enough information to influence the desired outcome. Initially, as a run progresses the information contained in the Seed is lost. When there are two possible ways to reach a value in a run we lose the information about which path was taken to reach it[2].
Odd numbers always transition (3*n + 1) to even numbers so an odd value can only be reached from one even value. However, you can reach certain even numbers from either an odd or even transition. For example, an output of 16 can be reached from either 32 or 5.
32 -> 32 / 2 = 16 5 -> 3*5 + 1 = 16
Whenever transitioning to an even value such that (even % 3 = 1) then the previous value could have been either:
2 * even or (even - 1) / 3
For Collatz, a bit of information contained in the seed is lost each time one of these select even numbers is reached. After all bits in the seed are scrubbed this initial phase is complete. Any attempt to contrive a Seed to skew results can only directly affect values during this phase.
One way hash functions rely on this concept of lost information[3]. Secure hashes have very many ways to reach each hashed value. This is how passwords are encoded and used for authentication. Using this metaphor you can think of the Seed as a password and the Collatz sequence as a trivial one way hash schema used to mask it.
This next phase is key as this is where the sequence runs below the Seed. The sequence is rewritten as a pseudo random number (PRNG) generator. Hastad et al. (1999)[4] show that any one way hash can be used to create a PRNG. Uniformly randomized values eventually trend towards their average. In turn this drives transitions towards their average gain. In the introduction we showed that Collatz has an average gain of 0.86603; eventually driving the series below the Seed value.
We'll be using the the combined series from section 1.2 for each randomization step. The value, j, is always even so the [j + 1] term will simply set the low order bit to one as there is no carry. Also the product will have an odd result so that decrementing by 1 will likewise just clear the low order bit.
Input: ([j + 1] * 2^ko - 1) * 2^kz Result: [j + 1] * 3^ko - 1 = [j ⊻ 1] * 3^ko ⊻ 1
In this next example the top line has steps for a randomization phase that begins with 647. Calculations for each combined transition (1942, 2186, 1640, 308, 116) are shown in binary.
647 1942 971 2914 1457 4372 2186 1093 3280 1640 820 410 205 616 308 154 77 232 116
1942 2186 1640 116
Input 11110010_110 1000100010_10 1100110_1000 100110_100
Shift Right 11110010 1000100010 1100110 100110
Xor 1 11110011 1000100011 1100111 100111
Times 3^ko 100010001011 11001101001 100110101 1110101
Xor 1 1000100010_10 1100110_1000 100110_100 1110100
A pseudo random number generator repeatedly applies a function to produce a series of values. In order to produce uniformly random numbers, operators cannot be biased towards producing either more ones or zeros. In a uniform sequence the entropy will be one. If it is not uniform the bias will show up in the operators.
If you remove some low order bits of a random number, the remaining part will still be random. Using the upper bits from the input still gives random values. However, the way the value is split the selected upper value will be even. The low order bit is zero and only the other bits are randomized portion.
The first Exclusive Or sets the low bit of the selected region. This is balanced out by clearing it in the final step with another Exclusive.
The product of a random variable by a constant is also random, but with a larger gap between them. Multiplying random numbers from 1 to 10 by 3 yields random numbers from 3 to 33. They simply have a gap of 3 between them instead of 1.
The product used to scramble values is equivalent to repeated sums of the input. The following table shows all combinations for the three inputs (A, B, Carry In) and the two outputs (Sum, Carry Out). It also shows changes (Exclusive Or) between the sum and inputs A and B.
Carry Carry
A B In Sum Out A ⊻ Sum B ⊻ Sum
0 0 0 | 0 0 | 0 0
0 0 1 | 1 0 | 1 1
0 1 0 | 1 0 | 1 0
0 1 1 | 0 1 | 0 1
1 0 0 | 1 0 | 0 1
1 0 1 | 0 1 | 1 0
1 1 0 | 0 1 | 1 1
1 1 1 | 1 1 | 0 0
Input bits A and B are vertically aligned and are altered by addition. Carries are applied horizontally and propagate to higher order bits. This way bits in both directions become scrambled.
Note that all the columns in the table are different. This shows how bits are scrambled to produce randomized results. Also note that all columns contain 4 zeros and 4 ones. This balance produces results that are unbiased towards either zero or one bits. The end result is a series of uniformly distributed pseudo random numbers.
Outputs in any individual series depend on the values kz and ko. The more random they are then the more random the series. The k values measure the width of a horizontal subset of bits in each value. Pseudo random number generators that conflate operations on horizontal and vertical sets of bits rely on the independence of these orthogonal values.
The repeated zeros and upwardly ones in the lowest bits that might have low entropy are continuously removed and replaced with scrambled bits. This creates a self regulating system that continuosly randomizes the lower bits. Since those bits control the selection operation in the next round.
When runs have uniformly random values then revisiting the Syracuse sequence, the average number of even and odd transitions will be the balance out. In turn this causes the run to decrease since the average gain is less than one. If the sequence was not uniformly random then we would see bias amongst the arithmetic operations used in each round.
Examples where seeds produce long runs will have highly skewed values in the first phase, but that cannot be sustained. As values become more randomized and the series progresses they will trend toward average results. With coin tossing even if you get lucky and call the results of several coin tosses, your luck will run out in the long run.
The previous randomization phase leaves us with a value of N that is below the seed. It is well understood that once this happens we know the series will eventually terminate at one. Firstly, we know all values below some arbitrary small number M (say 10) transition to one.
Next, starting with the next higher Seed, M + 1, we transition until it reaches M or less, Since we already know Seeds of M or less will reach one, by induction, once a series goes below its Seed we know it will reach one. This is why even Seeds are uninteresting as they immediately decline.
When measuring the length of a run it including this phase is not useful and can distort any result. When winding down as numbers get smaller numbers they can become more irregular. Instead of defining the run length as the number of steps to one, use only the numer of odd steps until the series goes below the Seed. It is usually more practical to only count only odd steps because corresponds to the number of terms in the algebraic expansion of a run.
Now we will take some entropy measures to verify that the Randomization phase matches expectations. The first few values will have lower entropy until enough bits are included to average out. In the Introduction we've shown that to sustain an infinite run there needs to be 64% or more ones. This gives and entropy of:
H = -.36 * log2( .36 ) + -.64 * log2( .64 ) = 0.94268
Individual runs will typically have some jitter since we are performing discrete computations. There will be higher entropy at the end of very long runs; which are rare. In the first phase long runs will be skewed towards more odd steps in order to make the values grow larger up front. Short runs where evens dominate won't even make it to the randomization phase.
To compute the entropy the low order zero bit is discarded as it is fixed. Also, since the values have a variable width, the uppermost bit will be one is also discarded. THis differs from practical PRNG's where the values have a fixed width.
To see the randomization in action this trace lists entropy in the first two phases. Entropy is computed using the accumulated number of ones and zeros in the run. The counts of ones and zeros are reset at the start of the Randomization phase so that those computations are completely separate. Even in the Information Loss phase entropy is well above 0.94268 bound right out of the gate. The computed length of the Information Loss phase is quite conservative.
Seed = 4_50449_75045_09599 = #10_00d1_0da5_de9f
Iteration Entropy Ones Zeros
1 0.99750 24 27 Information Loss phase 1
2 0.99993 52 51
3 0.99988 77 79
4 0.99998 105 104
5 0.99934 136 128
6 0.99965 163 156
7 0.99950 194 184
8 0.99986 222 216
9 0.99999 250 248
10 0.99994 281 276
11 0.99973 314 302
12 0.99984 342 332
13 0.99993 369 362
14 0.99966 402 385
15 0.99983 428 415
16 0.99989 456 445
17 0.99982 487 472
18 0.99975 518 499
19 0.99988 545 531
20 0.99920 31 29 Randomization phase 2
30 1.00000 321 321
40 0.99978 606 585
50 0.99966 899 861
60 0.99983 1192 1156
70 0.99969 1489 1429
80 0.99996 1770 1744
90 0.99963 2105 2012
100 0.99967 2410 2309
110 0.99876 2772 2551
120 0.99828 3105 2816
130 0.99890 3387 3133
If the hash function had a bias it would show up by running it over many consecutive numbers. Here the hash was run over a million consecutive even numbers. This next chart shows the cumulative entropy of the resulting values; which is very near one as expected.
Iteration Entropy Ones Zeros 50_000 0.99350 475_206 574_794 100_000 0.99721 984_678 1_115_322 150_000 0.99886 1_512_405 1_637_595 200_000 0.99966 2_054_339 2_145_661 250_000 0.99984 2_585_879 2_664_121 300_000 0.99970 3_086_254 3_213_746 350_000 0.99987 3_625_941 3_724_059 400_000 0.99970 4_113_665 4_286_335 450_000 0.99976 4_638_681 4_811_319 500_000 0.99983 5_168_585 5_331_415 550_000 0.99991 5_711_156 5_838_844 600_000 0.99996 6_255_089 6_344_911 650_000 1.00000 6_816_718 6_833_282 700_000 0.99994 7_415_041 7_284_959 750_000 1.00000 7_882_886 7_867_114 800_000 1.00000 8_392_173 8_407_827 850_000 1.00000 8_925_576 8_924_424 900_000 1.00000 9_428_185 9_471_815 950_000 0.99999 9_944_542 10_005_458 1_000_000 0.99999 10_466_775 10_533_225 1_048_576 1.00000 11_012_891 11_007_205
This next chart shows a contorted sequence where some bits are artificially forced to one. This shows how skewed the operators have to get in order for entropy to drop below 0.94268 where it would increase indefinitely. 11 out of 20 bits are needed to be forced to one to sufficiently skew the result.
Value Of Final Forced Bits Entropy Ones Zeros 0 1.00000 11_012_891 11_007_205 #10_000 0.99995 11_102_306 10_917_790 #11_000 0.99971 11_229_280 10_790_816 #11_100 0.99931 11_350_975 10_669_121 #11_500 0.99916 11_385_687 10_634_409 #15_500 0.99892 11_435_329 10_584_767 #55_500 0.99703 11_716_482 10_303_614 #155_500 0.99727 11_687_638 10_332_458 #175_500 0.98541 12_573_465 9_446_631 #177_500 0.96792 13_323_124 8_696_972 #177_700 0.95400 13_775_529 8_244_567 #177_740 0.94266 14_093_550 7_926_546
The Collatz sequence incorporates principle mechanisms commonly used to create pseudo random number generators.
Any individual run is partitioned into three phases. In the initial phase the Seed value can influence the outcome to produce arbitrarily long runs. After that the series generates randomized values until it goes below the Seed value. From there it is guaranteed to reduce to one unless the series is circular.
A uniformly randomized series eventually moves towards a statistically average gain. For a Collatz series to sustain an average gain above one would require over 1.7 times more odd transitions than even. This is well above parity. Instead, randomization forces the series to average out and decrease until it inevitably goes below the seed. Once it does that we know it will terminate.
The random behavior of the Collatz sequence makes it impossible to prove algebraically. Conway[5] showed that a generalization of the 3*N + 1 problem is undecidable. Trying to make sense of the values in the series is akin to analysing values produced by a random number generator. The irony is that this randomness is the force that leads to convergence.
[1] Behrouz Zolfaghari, Khodakhast Bibak, and Takeshi Koshiba
"The Odyssey of Entropy: Cryptography"; Entropy 2022, 24(2), 266.
https://www.mdpi.com/1099-4300/24/2/266/pdf
[2] John C. Baez, Tobias Fritz, Tom Leinster,
"A Characterization of Entropy in Terms of Information Loss",
Entropy 2011, 13(11), 1945-1957
https://mdpi-res.com/d_attachment/entropy/entropy-13-01945/article_deploy/entropy-13-01945-v2.pdf
[3] Russell Impagliazzo, Leonid A. Levin, Michael Luby;
"Pseudo-random Generation from one-way functions", in David S. Johnson (ed.),
Proceedings of the 21st Annual ACM Symposium on Theory of Computing,
May 14-17, 1989, Seattle, Washington, USA, {ACM}, pp. 12-24,
doi:10.1145/73007.73009, S2CID 18587852
https://dl.acm.org/doi/pdf/10.1145/73007.73009
[4] Johan Hastad, Russell Impagliazzo, Leonid A. Levin, Michael Luby;
A pseudorandom generator from any one-way function.
Siam Journal on Computation 28(4):1364-1396, 1999.
https://epubs.siam.org/doi/10.1137/S0097539793244708
[5] John H. Conway, "Unpredictable iterations". Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49-52.
Here are the entropy values for the randomization phase of some long runs. The Sample Size is the number of values produced by the algorithm for randomization. All of the entropy values are well above 0.94268; the entropy required to produce an infinitely long run.
Entropy is computed from values in each run. Since at the end of each step the values are all even, the low order bit is discarded. Values also have variable widths; unlike values in a practical PRNG. To account for this the upper one bit is also discarded. The entropy is then computed using the total number of ones and zeros in each run. To illustrate this the chart shows the entropy after five steps into the randomization phase.
Run Sample Run Sample Length Seed Entropy Size Length Seed Entropy Size 200 371_871_359 0.99979 68 | 292 331_224_689_767 0.99971 117 201 247_914_239 0.99979 69 | 293 188_890_883_743 0.99946 92 202 165_276_159 0.99980 69 | 294 215_384_833_215 0.99991 104 203 2_173_615_775 0.99998 61 | 295 167_903_007_771 0.99952 92 204 293_824_283 0.99974 69 | 296 144_460_775_535 0.99995 104 205 195_882_855 0.99980 70 | 297 125_291_645_607 0.99938 101 206 620_752_511 0.99903 62 | 298 196_281_297_639 0.99975 118 207 348_236_187 0.99969 70 | 299 1_063_641_582_407 0.99992 102 208 413_835_007 0.99871 62 | 300 709_094_388_271 0.99993 102 209 1_651_171_495 0.99810 68 | 301 473_644_547_375 0.99953 103 210 127_456_255 0.99983 95 | 302 380_103_773_863 0.99979 103 211 245_235_559 0.99882 65 | 303 315_763_031_583 0.99956 104 212 5_425_672_039 1.00000 71 | 304 284_396_952_295 0.99929 105 213 217_987_163 0.99892 66 | 305 33_980_539_439 0.99951 103 214 290_649_551 0.99894 66 | 306 150_164_453_871 0.99984 105 215 193_766_367 0.99899 67 | 307 22_653_692_959 0.99946 105 216 145_324_775 0.99894 66 | 308 224_708_703_047 0.99948 109 217 96_883_183 0.99899 67 | 309 20_136_615_963 0.99955 106 218 1_583_507_967 0.99875 79 | 310 199_741_069_375 0.99937 110 219 661_398_811 0.99972 73 | 311 1_150_284_049_727 0.99959 104 220 1_968_165_887 0.99995 82 | 312 23_865_618_919 0.99942 107 221 2_079_441_767 0.99964 75 | 313 620_398_672_495 0.99999 108 222 326_610_023 0.99905 88 | 314 21_213_883_483 0.99963 108 223 984_082_943 0.99993 83 | 315 149_805_802_031 0.99934 109 224 1_232_261_787 0.99961 75 | 316 766_856_033_151 0.99950 105 225 656_055_295 0.99993 86 | 317 1_131_779_353_631 0.99986 112 226 2_848_461_311 0.99983 68 | 318 99_870_534_687 0.99937 110 227 409_344_047 0.99999 90 | 319 478_337_265_823 0.99977 114 228 272_896_031 0.99999 92 | 320 566_918_240_975 0.99973 115 229 181_930_687 1.00000 93 | 321 425_188_680_731 0.99971 115 230 1_304_621_055 0.99981 83 | 322 140_284_537_063 0.99932 112 231 1_324_921_887 0.99734 73 | 323 188_972_746_991 0.99974 121 232 95_592_191 0.99992 95 | 324 251_963_662_655 0.99971 117 233 1_104_180_463 1.00000 82 | 325 167_975_775_103 0.99973 123 234 5_328_487_839 0.99997 80 | 326 3_654_218_733_311 0.99990 113 235 13_551_207_911 0.99998 82 | 327 125_981_831_327 0.99971 122 236 9_034_138_607 0.99997 82 | 328 566_619_806_719 0.99980 122 237 63_728_127 0.99984 96 | 329 2_253_835_349_759 0.99974 105 238 15_218_280_607 1.00000 75 | 330 1_325_730_144_347 0.99997 115 239 17_108_656_891 0.99890 84 | 331 1_649_531_356_143 0.99990 108 240 3_246_339_311 0.99929 80 | 332 1_176_549_020_911 0.99970 122 241 2_164_226_207 0.99921 80 | 333 1_287_402_586_111 0.99945 108 242 1_442_817_471 0.99917 81 | 334 26_130_934_783 0.99950 114 243 20_445_954_119 0.99967 73 | 335 1_303_333_417_199 0.99990 109 244 13_630_636_079 0.99952 73 | 336 83_987_887_551 0.99973 123 245 9_087_090_719 0.99952 73 | 337 881_989_193_575 0.99998 117 246 6_058_060_479 0.99950 74 | 338 20_646_664_519 0.99957 115 247 18_019_682_047 0.99994 90 | 339 1_267_630_141_951 0.99994 116 248 17_825_084_863 0.99996 86 | 340 18_352_590_683 0.99958 116 249 217_740_015 0.99914 90 | 341 24_470_120_911 0.99963 116 250 1_801_487_687 0.99891 83 | 342 883_820_096_231 0.99998 115 251 1_200_991_791 0.99876 84 | 343 21_751_218_587 0.99983 118 252 16_670_963_135 0.99965 75 | 344 12_235_060_455 0.99973 119 253 6_250_517_663 0.99957 93 | 345 5_086_317_509_375 0.99979 127 254 32_060_507_419 0.99952 86 | 346 898_696_369_947 0.99921 121 255 14_884_335_615 0.99982 92 | 347 18_570_171_467_519 0.99934 122 256 87_147_171_839 0.99957 89 | 348 5_157_142_856_607 0.99805 118 257 6_216_083_103 0.99957 81 | 349 4_018_818_772_839 0.99979 129 258 8_781_412_679 0.99977 79 | 350 5_509_607_710_143 0.99996 117 259 24_083_989_231 0.99918 85 | 351 10_681_465_356_287 0.99811 119 260 23_962_604_007 1.00000 92 | 352 7_120_976_904_191 0.99807 120 261 21_407_990_427 0.99914 86 | 353 4_747_317_936_127 0.99800 125 262 5_854_275_119 0.99974 80 | 354 7_244_052_517_375 0.99801 118 263 3_902_850_079 0.99960 82 | 355 19_754_675_554_139 0.99926 120 264 349_414_071_423 1.00000 89 | 356 13_169_783_702_759 0.99934 120 265 25_244_554_015 1.00000 94 | 357 17_559_711_603_679 0.99914 120 266 81_774_557_807 0.99981 90 | 358 12_380_114_311_679 0.99926 122 267 60_142_063_643 0.99941 88 | 359 8_253_409_541_119 0.99938 122 268 4_111_644_527 0.99947 84 | 360 18_026_976_767_615 0.99973 129 269 5_482_192_703 0.99940 83 | 361 5_521_395_748_159 0.99990 124 270 2_741_096_351 0.99950 86 | 362 8_011_989_674_495 0.99974 130 271 23_759_827_611 0.99944 89 | 363 4_141_046_811_119 0.99994 125 272 139_869_168_255 0.99948 94 | 364 7_121_768_599_551 0.99972 132 273 1_827_397_567 0.99953 86 | 365 31_508_135_471_707 0.99968 130 274 28_159_795_687 0.99936 89 | 366 2_760_697_874_079 0.99996 126 275 59_834_174_399 0.99830 92 | 367 33_508_530_061_951 0.99998 119 276 4_704_765_167 0.99948 96 | 368 18_307_067_699_951 0.99611 119 277 6_273_020_223 0.99944 96 | 369 12_204_711_799_967 0.99612 120 278 39_889_449_599 0.99845 94 | 370 2_813_538_212_167 0.99983 137 279 53_185_932_799 0.99837 94 | 371 23_838_942_284_287 0.99939 129 280 3_136_510_111 0.99944 96 | 372 2_500_922_855_259 0.99980 138 281 85_153_952_959 0.99926 92 | 373 28_729_623_136_495 0.99997 125 282 2_788_008_987 0.99946 96 | 374 22_686_053_577_471 0.99917 133 283 23_440_029_087 0.99930 94 | 375 25_537_442_787_995 0.99997 125 284 338_669_420_543 0.99909 93 | 376 17_024_961_858_663 0.99995 125 285 37_354_180_511 0.99850 96 | 377 42_992_933_789_863 0.99999 133 286 49_805_574_015 0.99846 96 | 378 30_750_102_727_359 0.99967 124 287 24_902_787_007 0.99846 97 | 379 89_394_821_046_783 0.99935 141 288 52_261_869_567 0.99959 112 | 380 3_122_627_652_839 0.99990 139 289 75_259_871_231 0.99921 97 | 381 2_081_751_768_559 0.99988 140 290 50_173_247_487 0.99918 97 | 382 67_321_861_993_087 0.99731 133 291 153_335_491_615 0.99991 102 | 383 70_558_238_740_647 0.99904 137