The table on the left shows the first 38 residue records, ignoring multiples of three, found by searching up to 4 billion. The tree shows how they are connected. The values after number 29 I calculated using C++ and Microsoft's 64 bit integers. I start worrying about the accuracy towards the end, 64 bit iterations might start to overflow around 4 billion and the floating point maths to calculate the residues gets a bit inaccurate.
I started using the equation 2(O-E)/(1.5E * N) to limit the size of the numerator and denominator
| Position | N | E | O | R(N) | Delta |
|---|---|---|---|---|---|
| 1 | 1 | 1.0 | |||
| 2 | 5 | 1 | 4 | 1.0666666666666667 | 0.0667 |
| 3 | 7 | 5 | 11 | 1.2039976484420929 | 0.1373 |
| 4 | 37 | 6 | 15 | 1.2148444741037334 | 0.01085 |
| 5 | 43 | 9 | 20 | 1.2389111604985532 | 0.02407 |
| 6 | 203 | 12 | 27 | 1.2441100214165366 | 0.005199 |
| 7 | 379 | 19 | 39 | 1.2480350469705168 | 0.003925 |
| 8 | 505 | 20 | 41 | 1.2488588324800682 | 0.0008238 |
| 9 | 559 | 30 | 57 | 1.2521613757221293 | 0.003303 |
| 10 | 745 | 31 | 59 | 1.2527216268969488 | 0.0005603 |
| 11 | 3,973 | 32 | 63 | 1.2528267298105236 | 0.0001051 |
| 12 | 5,297 | 33 | 65 | 1.2529055685701871 | 7.884e-005 |
| 13 | 14,125 | 34 | 68 | 1.2529351356632505 | 2.957e-005 |
| 14 | 18,833 | 35 | 70 | 1.2529573118988595 | 2.218e-005 |
| 15 | 44,641 | 38 | 76 | 1.2529625095676353 | 5.198e-006 |
| 16 | 50,221 | 36 | 73 | 1.2529656281896004 | 3.119e-006 |
| 17 | 52,907 | 41 | 81 | 1.2529826834280713 | 1.706e-005 |
| 18 | 141,085 | 42 | 84 | 1.2529856437774669 | 2.960e-006 |
| 19 | 188,113 | 43 | 86 | 1.2529878640486936 | 2.220e-006 |
| 20 | 250,817 | 44 | 88 | 1.2529895292572772 | 1.665e-006 |
| 21 | 594,529 | 47 | 94 | 1.2529899195411793 | 3.903e-007 |
| 22 | 626,335 | 52 | 102 | 1.2529900759597177 | 1.564e-007 |
| 23 | 668,845 | 45 | 91 | 1.2529901537116372 | 7.775e-008 |
| 24 | 891,793 | 46 | 93 | 1.2529906220528158 | 4.683e-007 |
| 25 | 1,189,057 | 47 | 95 | 1.2529909733089293 | 3.513e-007 |
| 26 | 1,585,409 | 48 | 97 | 1.2529912367511438 | 2.634e-007 |
| 27 | 1,879,003 | 51 | 102 | 1.2529914096351573 | 1.729e-007 |
| 28 | 2,505,337 | 52 | 104 | 1.2529915763447868 | 1.667e-007 |
| 29 | 6,590,975 | 65 | 126 | 1.2529918107138616 | 2.344e-007 |
| 30 | 15,623,051 | 68 | 132 | 1.252992 | |
| 31 | 32,917,703 | 73 | 141 | 1.252992 | |
| 32 | 87,780,541 | 74 | 144 | 1.252992 | |
| 33 | 117,040,721 | 75 | 146 | 1.252992 | |
| 34 | 230,930,791 | 87 | 166 | 1.252992 | |
| 35 | 328,805,755 | 81 | 157 | 1.252992 | |
| 36 | 512,600,731 | 97 | 183 | 1.252992 | |
| 37 | 2,733,870,565 | 98 | 187 | 1.252992 | |
| 38 | 3,240,142,891 | 101 | 192 | 1.252992 | |
| ? | 17,280,762,085 | 1.25299194097689 | |||
| 23,041,016,113 | 1.25299194099502 | ||||
| 30,721,354,817 | 1.25299194100861 | ||||
| 81,923,612,845 | 1.25299194101371 | ||||
| 95,787,794,023 | 1.25299194104084 |
The Tree
This tree shows how these values are connected and where the reverse route deviated from taking the smallest available value. The numbers in red are multiples of three and will have larger residue values than those in the table.
