Computer proof that completely multiplicative sequences have discrepancy greater than 2
The following C code implements this algorithm for making deductions about multiplicative functions of discrepancy 2. At any given time, the "knowledge" that this algorithm has is given by:
- a reduced expression for f(n) for each n, of the form f(n) = f(m) for some |m| \leq n (where we adopt the convention that f is odd, i.e. f(-m) = -f(m)). Thus for instance if we know that f(2)=-1, then we can reduce f(12)=f(-3). This is recorded by an array vals, thus for instance vals[12]=-3 would be recording the fact that f(12)=f(-3).
- Upper and lower bounds for each of the partial sums f[1,n] := f(1)+...+f(n).
The "easy" deductions that the algorithm makes automatically are:
- consequences of multiplicativity (e.g. f(n^2)=1);
- Adjusting upper and lower bounds for f[1,n] based on knowledge of f[1,n-1] and f(n), or of f[1,n+1] and f(n+1);
- Deducing the value of f(n) given enough information about f[1,n] and f[1,n-1]
The format of the program is
- [program name] [math]\displaystyle{ n_1 }[/math] [math]\displaystyle{ n_2 }[/math] ...
which makes the assumptions [math]\displaystyle{ f(n_1)=f(n_2)=\ldots=1 }[/math] and sees what one can deduce from there. For instance, "[program name] 2 -3" would assume that f(2)=1 and f(3)=-1.
Currently the computer only looks at f(n) for n < SIZE, where SIZE by default is 400. But one could certainly enlarge SIZE which in principle would make the computer "smarter".
Below are some examples of how this code works.
With no assumptions
[program name]
Deducing... Deducing...
| 0 | 1 2 3 1 5 6 7 2 1 [0-9]
10 11 3 13 14 15 1 17 2 19 [10-19]
5 21 22 23 6 1 26 3 7 29 [20-29]
30 31 2 33 34 35 1 37 38 39 [30-39]
10 41 42 43 11 5 46 47 3 1 [40-49]
2 51 13 53 6 55 14 57 58 59 [50-59]
15 61 62 7 1 65 66 67 17 69 [60-69]
70 71 2 73 74 3 19 77 78 79 [70-79]
5 1 82 83 21 85 86 87 22 89 [80-89]
10 91 23 93 94 95 6 97 2 11 [90-99]
1 101 102 103 26 105 106 107 3 109 [100-109]
110 111 7 113 114 115 29 13 118 119 [110-119]
30 1 122 123 31 5 14 127 2 129 [120-129]
130 131 33 133 134 15 34 137 138 139 [130-139]
35 141 142 143 1 145 146 3 37 149 [140-149]
6 151 38 17 154 155 39 157 158 159 [150-159]
10 161 2 163 41 165 166 167 42 1 [160-169]
170 19 43 173 174 7 11 177 178 179 [170-179]
5 181 182 183 46 185 186 187 47 21 [180-189]
190 191 3 193 194 195 1 197 22 199 [190-199]
Assuming f(2)=+1
[program name] 2
Setting f(2)=1... Deducing... Setting f(3)=-1... Deducing... Setting f(5)=-1... Deducing... Setting f(7)=-1... Deducing... Deducing...
| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 11 -1 13 -1 1 1 17 1 19 [10-19]
-1 1 11 23 -1 1 13 -1 -1 29 [20-29]
1 31 1 -11 17 1 1 37 19 -13 [30-39]
-1 41 1 43 11 -1 23 47 -1 1 [40-49]
1 -17 13 53 -1 -11 -1 -19 29 59 [50-59]
1 61 31 -1 1 -13 -11 67 17 -23 [60-69]
1 71 1 73 37 -1 19 -11 -13 79 [70-79]
-1 1 41 83 1 -17 43 -29 11 89 [80-89]
-1 -13 23 -31 47 -19 -1 97 1 11 [90-99]
1 101 -17 103 13 -1 53 107 -1 109 [100-109]
-11 -37 -1 113 -19 -23 29 13 59 -17 [110-119]
1 | 1 61 -41 31 -1 -1 127 1 -43 [120-129]
-13 131 -11 -19 67 1 17 137 -23 139 [130-139]
1 -47 71 143 1 -29 73 -1 37 149 [140-149]
-1 151 19 17 -11 -31 -13 157 79 -53 [150-159]
-1 -23 1 163 41 11 83 167 1 | 1 [160-169]
-17 19 43 173 -29 -1 11 -59 89 179 [170-179]
-1 181 -13 -61 23 -37 -31 187 47 1 [180-189]
-19 191 -1 193 97 13 1 197 11 199 [190-199]
Note that f[15,19] = 3 + f(17)+f(19) and f[168,171]=2-f(17)+f(19), giving a nontrivial deduction that f(19)=-1. We insert this information:
[program name] 2 -19
Setting f(2)=1... Setting f(19)=-1... Deducing... Setting f(3)=-1... Deducing... Setting f(5)=-1... Deducing... Setting f(7)=-1... Deducing... Deducing...
| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 11 -1 13 -1 1 1 17 1 -1 [10-19]
-1 1 11 23 -1 1 13 -1 -1 29 [20-29]
1 31 1 -11 17 1 1 37 -1 -13 [30-39]
-1 41 1 43 11 -1 23 47 -1 1 [40-49]
1 -17 13 53 -1 -11 -1 1 29 59 [50-59]
1 61 31 -1 1 -13 -11 67 17 -23 [60-69]
1 71 1 73 37 -1 -1 -11 -13 79 [70-79]
-1 1 41 83 1 -17 43 -29 11 89 [80-89]
-1 -13 23 -31 47 1 -1 97 1 11 [90-99]
1 101 -17 103 13 -1 53 107 -1 109 [100-109]
-11 -37 -1 113 1 -23 29 13 59 -17 [110-119]
1 | 1 61 -41 31 -1 -1 127 1 -43 [120-129]
-13 131 -11 1 67 1 17 137 -23 139 [130-139]
1 -47 71 143 1 -29 73 -1 37 149 [140-149]
-1 151 -1 17 -11 -31 -13 157 79 -53 [150-159]
-1 -23 1 163 41 11 83 167 1 | 1 [160-169]
-17 -1 43 173 -29 -1 11 -59 89 179 [170-179]
-1 181 -13 -61 23 -37 -31 187 47 1 [180-189]
1 191 -1 193 97 13 1 197 11 199 [190-199]
Assuming f(2)=f(37)=+1
[program name] 2 -19 37
Setting f(2)=1... Setting f(19)=-1... Setting f(37)=1... Deducing... Setting f(3)=-1... Deducing... Setting f(5)=-1... Deducing... Setting f(7)=-1... Deducing... Setting f(17)=-1... Deducing... Setting f(11)=1... Setting f(59)=-1... Setting f(151)=1... Deducing... Setting f(31)=-1... Setting f(29)=1... Setting f(101)=-1... Setting f(109)=1... Setting f(113)=1... Deducing... Setting f(61)=1... Setting f(103)=-1... Setting f(13)=-1... Setting f(127)=1... Deducing... Setting f(23)=1... Setting f(53)=1... Setting f(41)=1... Deducing... Setting f(43)=-1... Setting f(47)=-1... Setting f(67)=1... Setting f(83)=-1... Setting f(89)=-1... Setting f(107)=1... Setting f(79)=1... Setting f(163)=-1... Deducing... Contradiction! -1 >= f[1,47] >= 1
| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | 1 -1 | -1 -1 v 1 1 | -1 1 | -1 [10-19]
-1 v 1 1 | 1 -1 | 1 -1 | -1 -1 v 1 [20-29]
1 | -1 1 | -1 -1 v 1 1 | 1 -1 | 1 [30-39]
-1 | 1 1 ^ -1 1 ^ -1 1 ^ -1 -1 v 1 [40-49]
1 | 1 -1 | 1 -1 | -1 -1 v 1 1 | -1 [50-59]
1 | 1 -1 | -1 1 | 1 -1 | 1 -1 | -1 [60-69]
1 | 71 1 73 1 ^ -1 -1 | -1 1 | 1 [70-79]
-1 1 1 -1 1 | 1 -1 -1 1 -1 [80-89]
-1 v 1 1 | 1 -1 1 -1 97 1 | 1 [90-99]
1 ^ -1 1 ^ -1 -1 | -1 1 | 1 -1 | 1 [100-109]
-1 | -1 -1 v 1 1 | -1 1 | -1 -1 v 1 [110-119]
1 | 1 1 ^ -1 -1 | -1 -1 v 1 1 | 1 [120-129]
1 131 -1 1 1 1 -1 137 -1 139 [130-139]
1 1 71 -1 1 -1 73 -1 1 149 [140-149]
-1 | 1 -1 | -1 -1 v 1 1 | 157 1 ^ -1 [150-159]
-1 | -1 1 | -1 1 | 1 -1 167 1 | 1 [160-169]
1 ^ -1 -1 173 -1 | -1 1 | 1 -1 179 [170-179]
-1 181 1 -1 1 -1 1 -1 -1 1 [180-189]
1 191 -1 193 97 -1 1 197 1 199 [190-199]
Assuming f(2)=+1; f(37)=-1
[program name] 2 -19 37
Setting f(2)=1... Setting f(19)=-1... Setting f(37)=-1... Deducing... Setting f(3)=-1... Deducing... Setting f(5)=-1... Setting f(77)=1... Deducing... Setting f(7)=-1... f(11)=f(77)/f(11)=-1... Setting f(11)=-1... Deducing... Setting f(13)=1... Setting f(31)=-1... Setting f(17)=-1... Setting f(67)=-1... Deducing... Setting f(23)=1... Setting f(41)=1... Setting f(53)=-1... Contradiction! f(1) = 1 and f(1) = -1 (from f(69)=1)
| 0 | 1 1 ^ -1 1 ^ -1 -1 | -1 1 | 1 [0-9]
-1 | -1 -1 v 1 -1 v 1 1 | -1 1 | -1 [10-19]
-1 v 1 -1 v 1 -1 1 1 -1 -1 29 [20-29]
1 | -1 1 | 1 -1 | 1 1 ^ -1 -1 | -1 [30-39]
-1 v 1 1 43 -1 | -1 1 47 -1 v 1 [40-49]
1 | 1 1 ^ -1 -1 1 -1 1 29 59 [50-59]
1 61 -1 | -1 1 | -1 1 | -1 -1 v -1 [60-69]
1 71 1 73 -1 | -1 -1 v 1 -1 v 79 [70-79]
-1 1 1 83 1 | 1 43 -29 -1 89 [80-89]
-1 | -1 1 1 47 1 -1 97 1 -1 [90-99]
1 101 1 103 1 -1 -1 107 -1 109 [100-109]
1 | 1 -1 | 113 1 -1 29 1 59 v 1 [110-119]
1 | 1 61 -1 -1 | -1 -1 v 127 1 -43 [120-129]
-1 131 1 | 1 -1 | 1 -1 | 137 -1 139 [130-139]
1 -47 71 -1 1 -29 73 -1 -1 149 [140-149]
-1 151 -1 | -1 1 | 1 -1 | 157 79 1 [150-159]
-1 -1 1 163 1 -1 83 167 1 | 1 [160-169]
1 ^ -1 43 173 -29 -1 -1 -59 89 179 [170-179]
-1 181 -1 -61 1 v 1 1 | 1 47 1 [180-189]
1 191 -1 193 97 1 1 197 -1 199 [190-199]
Using f(2)=-1
[program name] -2
Setting f(2)=-1... Deducing... Deducing...
| 0 | 1 -1 | 3 1 5 -3 7 -1 1 [0-9]
-5 11 3 13 -7 15 1 17 -1 19 [10-19]
5 21 -11 23 -3 1 -13 3 7 29 [20-29]
-15 31 -1 33 -17 35 1 37 -19 39 [30-39]
-5 41 -21 43 11 5 -23 47 3 1 [40-49]
-1 51 13 53 -3 55 -7 57 -29 59 [50-59]
15 61 -31 7 1 65 -33 67 17 69 [60-69]
-35 71 -1 73 -37 3 19 77 -39 79 [70-79]
5 1 -41 83 21 85 -43 87 -11 89 [80-89]
-5 91 23 93 -47 95 -3 97 -1 11 [90-99]
1 101 -51 103 -13 105 -53 107 3 109 [100-109]
-55 111 7 113 -57 115 29 13 -59 119 [110-119]
-15 1 -61 123 31 5 -7 127 -1 129 [120-129]
-65 131 33 133 -67 15 -17 137 -69 139 [130-139]
35 141 -71 143 1 145 -73 3 37 149 [140-149]
-3 151 -19 17 -77 155 39 157 -79 159 [150-159]
-5 161 -1 163 41 165 -83 167 -21 1 [160-169]
-85 19 43 173 -87 7 11 177 -89 179 [170-179]
5 181 -91 183 -23 185 -93 187 47 21 [180-189]
-95 191 3 193 -97 195 1 197 -11 199 [190-199]
Assuming f(5)=+1
[program name] -2 5
Setting f(2)=-1... Setting f(5)=1... Deducing... Setting f(3)=-1... Deducing... Setting f(7)=-1... Setting f(241)=1... Deducing... Deducing...
| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | 11 -1 13 1 -1 1 17 -1 19 [10-19]
1 | 1 -11 23 1 | 1 -13 -1 -1 29 [20-29]
1 31 -1 -11 -17 -1 1 37 -19 -13 [30-39]
-1 41 -1 43 11 1 -23 47 -1 1 [40-49]
-1 -17 13 53 1 11 1 -19 -29 59 [50-59]
-1 61 -31 -1 1 13 11 67 17 -23 [60-69]
1 71 -1 73 -37 -1 19 -11 13 79 [70-79]
1 | 1 -41 83 1 17 -43 -29 -11 89 [80-89]
-1 -13 23 -31 -47 19 1 97 -1 11 [90-99]
1 101 17 103 -13 1 -53 107 -1 109 [100-109]
-11 -37 -1 113 19 23 29 13 -59 -17 [110-119]
1 | 1 -61 -41 31 1 1 127 -1 -43 [120-129]
-13 131 -11 -19 -67 -1 -17 137 23 139 [130-139]
-1 -47 -71 143 1 29 -73 -1 37 149 [140-149]
1 151 -19 17 11 31 -13 157 -79 -53 [150-159]
-1 -23 -1 163 41 -11 -83 167 -1 1 [160-169]
-17 19 43 173 29 -1 11 -59 -89 179 [170-179]
1 181 13 -61 -23 37 31 187 47 1 [180-189]
-19 191 -1 193 -97 -13 1 197 -11 199 [190-199]
-1 -67 -101 -29 -17 41 -103 23 13 209 [200-209]
-1 211 53 -71 -107 43 1 -31 -109 -73 [210-219]
11 221 37 223 1 | 1 -113 227 -19 229 [220-229]
-23 11 -29 233 -13 47 59 -79 17 239 [230-239]
-1 | 1 -1 | -1 61 1 41 247 -31 -83 [240-249]
-1 251 -1 253 -127 -17 1 257 43 -37 [250-259]
13 29 -131 263 11 53 19 -89 67 269 [260-269]
1 271 17 13 -137 11 -23 277 -139 31 [270-279]
1 281 47 283 71 -19 -143 -41 -1 1 [280-289]
-29 -97 73 293 1 59 -37 -11 -149 299 [290-299]
-1 -43 -151 -101 19 61 -17 307 -11 -103 [300-309]
-31 311 13 313 -157 -1 79 317 53 319 [310-319]
1 -107 23 323 1 13 -163 -109 -41 -47 [320-329]
11 331 83 37 -167 67 1 337 -1 -113 [330-339]
17 341 -19 -1 -43 -23 -173 347 -29 349 [340-349]
1 -13 -11 353 59 71 89 17 -179 359 [350-359]
-1 1 -181 -1 -13 73 61 367 23 41 [360-369]
-37 -53 -31 373 -187 -1 -47 377 -1 379 [370-379]
19 -127 -191 383 1 -11 -193 43 97 389 [380-389]
13 391 -1 -131 -197 79 11 397 -199 19 [390-399]
Assuming f(5)=f(11)= 1
[program name] -2 5 11
Setting f(2)=-1... Setting f(5)=1... Setting f(11)=1... Deducing... Setting f(3)=-1... Deducing... Setting f(7)=-1... Setting f(241)=1... Deducing... Setting f(23)=-1... Setting f(17)=-1... Setting f(19)=1... Deducing... Setting f(13)=-1... Setting f(47)=-1... Setting f(59)=1... Setting f(67)=-1... Setting f(239)=1... Setting f(251)=1... Setting f(31)=1... Deducing... Setting f(29)=-1... Setting f(53)=-1... Contradiction! f(1) = 1 and f(1) = -1 (from f(58)=-1)
| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | 1 -1 | -1 1 | -1 1 | -1 -1 v 1 [10-19]
1 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | -1 [20-29]
1 | 1 -1 | -1 1 | -1 1 | 37 -1 1 [30-39]
-1 41 -1 43 1 | 1 1 ^ -1 -1 | 1 [40-49]
-1 | 1 -1 | -1 1 | 1 1 ^ -1 1 1 [50-59]
-1 61 -1 | -1 1 | -1 1 | -1 -1 v 1 [60-69]
1 | 71 -1 73 -37 -1 1 -1 -1 79 [70-79]
1 | 1 -41 83 1 -1 -43 1 -1 89 [80-89]
-1 | 1 -1 | -1 1 | 1 1 ^ 97 -1 1 [90-99]
1 101 -1 103 1 | 1 1 107 -1 109 [100-109]
-1 -37 -1 113 1 -1 -1 | -1 -1 v 1 [110-119]
1 | 1 -61 -41 1 | 1 1 ^ 127 -1 -43 [120-129]
1 131 -1 | -1 1 | -1 1 | 137 -1 139 [130-139]
-1 1 -71 -1 1 -1 -73 -1 37 149 [140-149]
1 151 -1 | -1 1 | 1 1 ^ 157 -79 1 [150-159]
-1 1 -1 163 41 -1 -83 167 -1 v 1 [160-169]
1 | 1 43 173 -1 -1 1 -1 -89 179 [170-179]
1 181 -1 -61 1 37 1 -1 -1 1 [180-189]
-1 191 -1 193 -97 1 1 197 -1 199 [190-199]
-1 1 -101 1 1 41 -103 -1 -1 1 [200-209]
-1 211 -1 -71 -107 43 1 -1 -109 -73 [210-219]
1 | 1 37 223 1 | 1 -113 227 -1 229 [220-229]
1 | 1 1 233 1 -1 1 -79 -1 | 1 [230-239]
-1 | 1 -1 | -1 61 1 41 -1 -1 -83 [240-249]
-1 | 1 -1 | -1 -127 1 1 257 43 -37 [250-259]
-1 -1 -131 263 1 -1 1 -89 -1 269 [260-269]
1 271 -1 | -1 -137 1 1 277 -139 1 [270-279]
1 281 -1 283 71 -1 1 -41 -1 1 [280-289]
1 -97 73 293 1 | 1 -37 -1 -149 1 [290-299]
-1 -43 -151 -101 1 61 1 307 -1 -103 [300-309]
-1 311 -1 313 -157 -1 79 317 -1 -1 [310-319]
1 -107 -1 | -1 1 | -1 -163 -109 -41 1 [320-329]
1 331 83 37 -167 -1 1 337 -1 -113 [330-339]
-1 | 1 -1 | -1 -43 1 -173 347 1 349 [340-349]
1 | 1 -1 | 353 1 71 89 -1 -179 359 [350-359]
-1 1 -181 -1 1 73 61 367 -1 41 [360-369]
-37 1 -1 373 1 -1 1 1 -1 379 [370-379]
1 -127 -191 383 1 -1 -193 43 97 389 [380-389]
-1 1 -1 -131 -197 79 1 397 -199 1 [390-399]
Back to assuming f(5) = 1
From the previous section we can now assume f(11)=-1.
[program name] -2 5 -11
Setting f(2)=-1... Setting f(5)=1... Setting f(11)=-1... Deducing... Setting f(3)=-1... Deducing... Setting f(7)=-1... Setting f(13)=1... Setting f(23)=-1... Setting f(241)=1... Deducing... Contradiction! -1 >= f[1,23] >= 1
| 0 | 1 -1 | -1 1 | 1 1 ^ -1 -1 | 1 [0-9]
-1 | -1 -1 v 1 1 | -1 1 | 17 -1 19 [10-19]
1 1 1 ^ -1 1 | 1 -1 -1 -1 29 [20-29]
1 31 -1 1 -17 -1 1 37 -19 -1 [30-39]
-1 41 -1 43 -1 1 1 47 -1 1 [40-49]
-1 -17 1 53 1 -1 1 -19 -29 59 [50-59]
-1 61 -31 -1 1 1 -1 67 17 1 [60-69]
1 71 -1 73 -37 -1 19 1 1 79 [70-79]
1 | 1 -41 83 1 17 -43 -29 1 89 [80-89]
-1 -1 -1 -31 -47 19 1 97 -1 | -1 [90-99]
1 | 101 17 103 -1 1 -53 107 -1 109 [100-109]
1 -37 -1 113 19 -1 29 1 -59 -17 [110-119]
1 | 1 -61 -41 31 1 1 127 -1 -43 [120-129]
-1 131 1 -19 -67 -1 -17 137 -1 139 [130-139]
-1 -47 -71 -1 1 29 -73 -1 37 149 [140-149]
1 151 -19 17 -1 31 -1 157 -79 -53 [150-159]
-1 1 -1 163 41 1 -83 167 -1 1 [160-169]
-17 19 43 173 29 -1 -1 -59 -89 179 [170-179]
1 181 1 -61 1 37 31 -17 47 1 [180-189]
-19 191 -1 193 -97 -1 1 197 1 199 [190-199]
-1 -67 -101 -29 -17 41 -103 -1 1 -19 [200-209]
-1 211 53 -71 -107 43 1 -31 -109 -73 [210-219]
-1 17 37 223 1 1 -113 227 -19 229 [220-229]
1 -1 -29 233 -1 47 59 -79 17 239 [230-239]
-1 1 -1 | -1 61 1 41 19 -31 -83 [240-249]
-1 251 -1 1 -127 -17 1 257 43 -37 [250-259]
1 29 -131 263 -1 53 19 -89 67 269 [260-269]
1 271 17 1 -137 -1 1 277 -139 31 [270-279]
1 281 47 283 71 -19 1 -41 -1 1 [280-289]
-29 -97 73 293 1 59 -37 1 -149 -1 [290-299]
-1 -43 -151 -101 19 61 -17 307 1 -103 [300-309]
-31 311 1 313 -157 -1 79 317 53 -29 [310-319]
1 -107 -1 323 1 1 -163 -109 -41 -47 [320-329]
-1 331 83 37 -167 67 1 337 -1 -113 [330-339]
17 -31 -19 -1 -43 1 -173 347 -29 349 [340-349]
1 -1 1 353 59 71 89 17 -179 359 [350-359]
-1 1 -181 -1 -1 73 61 367 -1 41 [360-369]
-37 -53 -31 373 17 -1 -47 29 -1 379 [370-379]
19 -127 -191 383 1 1 -193 43 97 389 [380-389]
1 -17 -1 -131 -197 79 -1 397 -199 19 [390-399]
No assumptions again
[program name] -2 -5
Setting f(2)=-1... Setting f(5)=-1... Deducing... Deducing...
| 0 | 1 -1 | 3 1 -1 -3 7 -1 1 [0-9]
1 11 3 13 -7 -3 1 17 -1 19 [10-19]
-1 21 -11 23 -3 1 -13 3 7 29 [20-29]
3 31 -1 33 -17 -7 1 37 -19 39 [30-39]
1 41 -21 43 11 -1 -23 47 3 1 [40-49]
-1 51 13 53 -3 -11 -7 57 -29 59 [50-59]
-3 61 -31 7 1 -13 -33 67 17 69 [60-69]
7 71 -1 73 -37 3 19 77 -39 79 [70-79]
-1 1 -41 83 21 -17 -43 87 -11 89 [80-89]
1 91 23 93 -47 -19 -3 97 -1 11 [90-99]
1 101 -51 103 -13 -21 -53 107 3 109 [100-109]
11 111 7 113 -57 -23 29 13 -59 119 [110-119]
3 1 -61 123 31 -1 -7 127 -1 129 [120-129]
13 131 33 133 -67 -3 -17 137 -69 139 [130-139]
-7 141 -71 143 1 -29 -73 3 37 149 [140-149]
-3 151 -19 17 -77 -31 39 157 -79 159 [150-159]
1 161 -1 163 41 -33 -83 167 -21 1 [160-169]
17 19 43 173 -87 7 11 177 -89 179 [170-179]
-1 181 -91 183 -23 -37 -93 187 47 21 [180-189]
19 191 3 193 -97 -39 1 197 -11 199 [190-199]
-1 201 -101 203 51 -41 -103 23 13 209 [200-209]
21 211 53 213 -107 -43 -3 217 -109 219 [210-219]
-11 221 -111 223 -7 1 -113 227 57 229 [220-229]
23 231 -29 233 -13 -47 59 237 -119 239 [230-239]
-3 241 -1 3 61 -1 -123 247 -31 249 [240-249]
1 251 7 253 -127 -51 1 257 -129 259 [250-259]
-13 29 -131 263 -33 -53 -133 267 67 269 [260-269]
3 271 17 273 -137 11 69 277 -139 31 [270-279]
7 281 -141 283 71 -57 -143 287 -1 1 [280-289]
29 291 73 293 -3 -59 -37 33 -149 299 [290-299]
3 301 -151 303 19 -61 -17 307 77 309 [300-309]
31 311 -39 313 -157 -7 79 317 -159 319 [310-319]
-1 321 -161 323 1 13 -163 327 -41 329 [320-329]
33 331 83 37 -167 -67 21 337 -1 339 [330-339]
-17 341 -19 7 -43 -69 -173 347 87 349 [340-349]
-7 39 -11 353 -177 -71 89 357 -179 359 [350-359]
1 | 1 -181 3 91 -73 -183 367 23 41 [360-369]
37 371 93 373 -187 -3 -47 377 -21 379 [370-379]
-19 381 -191 383 -3 -77 -193 43 97 389 [380-389]
39 391 -1 393 -197 -79 11 397 -199 399 [390-399]
