[math-fun] Re: A sequence puzzle

1 Aug 2006

      Joseph Biberstine wrote:
...
For fellow decipherers, here are some basic diagnostics.
...
split into increasing runs
{1, 2, 6, 12, 60}, {3, 21, 168, 504, 2520, 27720}, {4, 52, 364, 5460,
21840, 371280, 1113840, 21162960}, {5, 105, 2310, 53130, 212520,
1062600, 13813800, 124324200}, {7, 203, 6090, 188790, 3020640, 33227040,
564859680}, {35, 1260, 46620, 885780, 11515140}
Within these runs, terms seem to rise exponentially.
In the first lot of numbers it seemed as if the records started with
1, 2, 3, 4, 5, ... when sorted as follows:

{ 1}
{ 2,     6,    12,     60}
{ 3,    21,   168,    504,     2520,    27720}
{ 4,    52,   364,   5460,    21840,   371280,   1113840,  21162960}
{ 5, ... }

but then the next lot didn't fit this idea:

{ 5,  105,  2310,  53130,   212520,  1062600,  13813800, 124324200}
{ 7,  203,  6090, 188790,  3020640, 33227040, 564859680}
{35, 1260, 46620, 885780, 11515140}
...
successive ratios
I also noticed the successive ratios being integer *within* the runs.
...
{2, 3, 2, 5, 1/20, 7, 8, 3, 5, 11, 1/6930, 13, 7, 15, 4, 17, 3, 19,
1/4232592, 21, 22, 23, 4, 5, 13, 9, 1/17760600, 29, 30, 31, 16, 11, 17,
1/16138848, 36, 37, 19, 13, 1/1151514}
The construction principle seems to be related to playing with primes
and I will write the above ratios within the runs emphasizing that.
Each run becomes a block, where the first line contains the primes
which are newest and biggest. The second line contains the other factors
which are either smaller primes or composite:

block 1: {  1}
         {   }

block 2: {  2    3           5}
         {             2      }

block 3: {       7                       11}
         {  3          8     3     5       }

block 4: {      13                       17           19}
         {  4         7     15     4            3       }

block 5: {                  23                          }
         {  5   21    22           4      5    13      9}

block 6: {      29          31                   }
         {  7         30          16     11    17}

block 7: {            37             }
         { 35   36          19     13}

================

I get some pleasing pattern by completing the upper lines in the blocks:

block 1: {  1}
         {   }

block 2: {  2    3     4     5}
         {             2      }

block 3: {  6    7     8     9    10     11}
         {  3          8     3     5       }

block 4: { 12   13    14    15    16     17    18     19}
         {  4          7    15     4            3       }

block 5: { 20   21    22    23    24     25    26     27}
         {  5   21    22           4      5    13      9}

block 6: { 28   29    30    31    32     33    34}
         {  7         30          16     11    17}

block 7: { 35   36    37    38     39}
         { 35   36          19     13}

As one can plainly see: the lower numbers are always a factor
of their upper neighbour! The given ratios don't look as
random as they did on first sight. I wanted to share this
with other "fellow decipherers" and feel encouraged by
Joseph's mail. I am quite far from a good explanation, but
this order looks good, doesn't it?

Best regards,
Rainer Rosenthal
r.rosenthal@web.de