3x+1 Residue Records

The table on the left shows the first 38 residue records (found by searching up to 4 billion). The tree on the left shows how they are connected. The tree also shows 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. The ones after 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 get a bit inaccurate.

I started using the equation 2^(O-E)/(1.5^E * 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

Working Backwards: The forward equation we introduced above is quite straight forward. This page explores working backward to find a number's predecessors.