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

[hv at crypt.org]

Now proposed as A254637, A254638.

(Given it's otherwise the same code as discussed below for A113917/8, maybe
this would be a better example for someone else to confirm.)

Hugo

Neil Sloane <njasloane at gmail.com> wrote:
:Hugo, Is what I meant to say...
:
:Hugo,   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 7:33 PM, Neil Sloane <njasloane at gmail.com> wrote:
:
:> 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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