[seqfan] Re: A113917 and A113918: zero-free squaring

[Neil Sloane]

Hans, certainly those two sequences are worth adding
to the OEIS! Please do so!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Tue, Feb 3, 2015 at 12:34 PM, <hv at crypt.org> wrote:

> 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
>
>
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