# Computer proof that completely multiplicative sequences have discrepancy greater than 2

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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] $n_1$ $n_2$ ...

which makes the assumptions $f(n_1)=f(n_2)=\ldots=1$ 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]