[seqfan] A113917 and A113918: zero-free squaring

[hv at crypt.org]

Back in Jan 2006, David Wilson introduced this question:

  For a number n, let f(n) be the set of numbers gotten by splitting n^2 at
  the 0 digits.  For example

  29648^2 = 879003904

  so f(29648) = { 4, 39, 879 }

  Let S be the smallest set of numbers containing 2 and fixed by f.  What is
  the largest element of S?

.. which eventually gave A113917 (largest element) and A113918 (cardinality
of the set).

I did say at the time "I don't have full confidence in the results", but
when trying to clean up my 2006 code recently, as part of a long-running
project to push all my maths code to Github, I found several bugs which
meant some of the results were wrong. Sorry about that.

I've fixed those and further improved the code, available under 'zerofree'
in <https://github.com/hvds/seq>, and will go update the sequences on
the assumption that my new code is correct. I'd still appreciate it if
someone could confirm some or all of the values though.

I estimate the cardinality for A113918(9) is between 10^10 and 10^13,
which I can't calculate with my current approach (but I have another
approach in mind that might reach it). Given the rate of growth, I think
n=10 (ie the original question) is likely to be beyond my means.

With the new code it's easy to change the calculation, and replacing
s -> s^2 with s -> 2s gives a new pair of sequences that grows slow enough
it's easy to calculate more terms; I'm not sure if they're also worth
adding to OEIS, or if there are different calculations that would also be
of interest.

Hugo
---
With calculation s -> s^2: "n: card(n) max(n)"
2: 2 2
3: 18 1849
4: 2 2
5: 3050 266423227914725931
6: 34762 3100840870711697060720215047
7: 3087549 845486430620513036335402848567278325780455810752216401
8: 2 4

With calculation s -> 2s: "n: card(n) max(n)"
2: 2 2
3: 6 16
4: 2 2
5: 20 192
6: 13 128
7: 72 32768
8: 3 4
9: 92 69632
10: 42 23552
11: 308 25722880
12: 34 425984
13: 900 717895680
14: 178 1051828224
15: 1739 217079873536
16: 4 8
17: 3349 2270641389568
18: 443 10603200512
19: 4523 156423849771008
20: 387 950175531008
21: 14364 25160124578398208
22: 1827 385584983965696
23: 18672 450589122059304960
24: 234 40722497536
25: 39426 53279734579488838656
26: 15882 127148822502119047168
27: 52664 299326717942059499520
28: 8858 43157851113903387312128
29: 128253 13526981441472537034752
30: 28346 449522648486053412864
31: 123087 371244129204723018366976
32: 5 16
33: 259207 23655711299608586448011264
34: 87797 103182870656711001112576
35: 363512 39823687474383259120435200
36: 44545 63973308447624725004288
37: 671389 313519863989706816307303809024
38: 182549 45225850656203876163438682112
39: 1336282 31933986316064959928909955072
40: 18049 6034750858947540643601186816
41: 1289210 4145806855637690163777954119680
42: 634402 143410752413726318705389116325888
43: 2679419 251570201273324198920857495653056512
44: 156629 9442738596003761319219036160
45: 3428818 288049927140258932406824739012608
46: 987498 20384387023837630566380055072075677696
47: 5876576 16579286652350303184601394767032483840
48: 13308 152556272234873601963528260943872
49: 8122478 463424116819682991065891465214793542008832
50: 3118809 63039435236897106221986787164071919616
51: 14743535 247126135557931098912701316497011638272
52: 796352 1083181655178944127338714024967634157568
53: 15921180 71286088956163866149580753955553592475648
54: 4952057 2150676694930424720837752491965232971776
55: 29170111 507477067298501219491044453334717130866688
56: 403964 1746977513106742264740052226757623808
57: 48839762 27320774362755367988623371083844092646391808
58: 11997500 6094365163190490383475585992644805477269504
59: 71116907 9188495794580645540482153719631435565136609280
60: 2675259 14463684581762047084433709884282673037312
61: 122451245 9107655699502841924691980127153411750800769679360
62: 17820844 199082515611433949561498862265902993781030912