[seqfan] yet more primes

[Chuck Seggelin]

> Others have sought prime lists, wherein one must always add the offset.
> (So there's no difference between population and depth.) There are two
> kinds of particular interest.
>
> 1) Cunningham chains:
> The multiplier is two, and the offset is either +1 or -1.
> So far, they've found:
>     2n-1, starting at 69257563144280941; length is 15 (see A005603,
A064812)
>     2n+1, starting at 95405042230542329; length is 14 (see A005602)
> When converted to trees (offset +-1), those starting points yield no new
> primes; but they're still deeper than (50n+-27, from 1879).

Thanks for pointing this out to me!  In the intervening days I've found 3
prime trees 15 generations deep:

    2p+-15 from 13
    2p+-4095 from 13043
    2p+-4095 from 2999

And 7 prime trees 14 generations deep:

    2p+-15 from 11
    2p+-255 from 19219
    2p+-4095 from 17093
    2p+-4095 from 10093
    2p+-4095 from 17659
    2p+-4095 from 30181
    4p+-63 from 47

I'm convinced that eventually I should be able to find a prime tree of depth
16, making it deeper than the record holder Cunningham Chain.

> 2) Arithmetic sequences of primes
> Here the multiplier is one. A record-holder is
>     1n+4609098694200, starting at 11410337850553; length is 22
>     (see http://mathworld.wolfram.com/PrimeArithmeticProgression.html )
> But it's silly to make a tree with a multiplier of one.

It seems as though there ought to be a prime tree out there somewhere deeper
than 22 generations.  Since every node in a prime tree can have up to two
children, whereas each "node" in a Cunningham Chain or arithmetic
progression can have only one child, it appears that the odds are in the
favor of prime trees for greater depth.  (Though I have no idea where to
start looking for trees of such depth.)

I've also found some remarkably wide prime trees in the last couple days.
The previous record was 6p+-5 from 2 which had a population of 28 nodes.
2p+-15 from 13 has a burgeoning population of 71 nodes!  (I have
affectionately nicknamed 2p+-15 from 13 "the monster" for this reason.)
Other "fat" trees discovered recently are:

    56 nodes: 2p+-15 from 11
    46 nodes: 4p+-4095 from 1237
    38 nodes: 3p+-80 from 37
    38 nodes: 6p+-35 from 179
    30 nodes: 5p+-18 from 5

I have concluded that it was a mistake to hold off exploring trees where the
offset (d) exceeded the coefficient (c).  I determined that my reasoning
that such cases were duplicative was erroneous.  I have found that trees
where d = c raised to some power minus 1 tend to be very generative.

Happy Holidays everyone!

                -- Chuck Seggelin

PS: Here's the 15-generation, 71-node "monster" (2p+-15 from 13) for the
curious:

+-----------------------------------------------------------------
|[13]
|  - = [11]
|  .   - = [7]
|  .   .   + = [29]
|  .   .       - = [43]
|  .   .       .   - = [71]
|  .   .       .   .   - = [127]
|  .   .       .   .   .   - = [239]
|  .   .       .   .   .   .   - = [463]
|  .   .       .   .   .   .       - = [911]
|  .   .       .   .   .   .       + = [941]
|  .   .       .   .   .   .           - = [1867]
|  .   .       .   .   .   .               - = [3719]
|  .   .       .   .   .   + = [269]
|  .   .       .   .   .       - = [523]
|  .   .       .   .   .           - = [1031]
|  .   .       .   .   .           + = [1061]
|  .   .       .   .   .               + = [2137]
|  .   .       .   .   .                   - = [4259]
|  .   .       .   .   .                   + = [4289]
|  .   .       .   .   .                       - = [8563]
|  .   .       .   .   + = [157]
|  .   .       .   + = [101]
|  .   .       + = [73]
|  .   .           - = [131]
|  .   .               + = [277]
|  .   .                   + = [569]
|  .   .                       - = [1123]
|  .   .                       + = [1153]
|  .   + = [37]
|  .       - = [59]
|  .       .   - = [103]
|  .       .       - = [191]
|  .       .           - = [367]
|  .       .           .   - = [719]
|  .       .           .       - = [1423]
|  .       .           .       .   + = [2861]
|  .       .           .       .       + = [5737]
|  .       .           .       .           + = [11489]
|  .       .           .       .               - = [22963]
|  .       .           .       .               + = [22993]
|  .       .           .       .                   - = [45971]
|  .       .           .       .                       + = [91957]
|  .       .           .       + = [1453]
|  .       .           + = [397]
|  .       .               + = [809]
|  .       + = [89]
|  .           - = [163]
|  .           .   - = [311]
|  .           .       - = [607]
|  .           .           + = [1229]
|  .           .               + = [2473]
|  .           .                   - = [4931]
|  .           + = [193]
|  .               + = [401]
|  .                   - = [787]
|  .                       - = [1559]
|  + = [41]
|      - = [67]
|      .   + = [149]
|      .       - = [283]
|      .       + = [313]
|      .           + = [641]
|      .               + = [1297]
|      .                   - = [2579]
|      .                   + = [2609]
|      .                       + = [5233]
|      + = [97]
|          - = [179]
|              + = [373]
|                  + = [761]
+-----------------------------------------------------------------

Ordered list of nodes in 2p+-15 from 13:
7, 11, 13, 29, 37, 41, 43, 59, 67, 71, 73, 89, 97, 101, 103, 127, 131, 149,
157, 163, 179, 191, 193, 239, 269, 277, 283, 311, 313, 367, 373, 397, 401,
463, 523, 569, 607, 641, 719, 761, 787, 809, 911, 941, 1031, 1061, 1123,
1153, 1229, 1297, 1423, 1453, 1559, 1867, 2137, 2473, 2579, 2609, 2861,
3719, 4259, 4289, 4931, 5233, 5737, 8563, 11489, 22963, 22993, 45971, 91957