[seqfan] Re: Some Remarkable Series for A193543(n), A193544(n)

[Sean A. Irvine]

Interesting. I don't have a formula for you, but I note that these
sequences have high numbers of small factors, perhaps there is some
clue here.

 > Factorization(548619947325464115417997303889817577847783424);
[ <2, 34>, <3, 12>, <7, 4>, <11, 3>, <19, 1>, <23, 1>, <31, 1>, <1153, 1>, <1203799147044001, 1> ]
 > Factorization(1608709703391986198862095660055287572797969137664);
[ <2, 36>, <3, 18>, <7, 7>, <11, 3>, <19, 2>, <23, 1>, <31, 1>, <73, 1>, <10853, 1>, <270322081, 1> ]
 > Factorization(5263789873527529111801402355636596520955646052401152);
[ <2, 38>, <3, 14>, <7, 5>, <11, 3>, <17, 1>, <19, 2>, <23, 1>, <31, 1>, <37952413, 1>, <1077724711057, 1> ]
 > Factorization(19109911894872035157002147075069564951191053217428406272);
[ <2, 40>, <3, 14>, <7, 5>, <11, 5>, <19, 2>, <23, 1>, <31, 1>, <9281, 1>, <561973920620386973, 1> ]

Similarly,

 > Factorization(387932885047590464537298997011364686273183744);
[ <2, 50>, <3, 12>, <7, 4>, <11, 3>, <19, 1>, <23, 1>, <31, 1>, <271, 1>, <55261048303, 1> ]
 > Factorization(1137529540229602302657361974173461626864204251136);
[ <2, 53>, <3, 13>, <7, 5>, <11, 3>, <19, 1>, <23, 1>, <31, 1>, <181, 1>, <1367, 1>, <10343, 1>, <102139, 1> ]
 > Factorization(3722061514312152863902279948170443229080820784300032);
[ <2, 56>, <3, 14>, <7, 5>, <11, 3>, <19, 2>, <23, 1>, <31, 1>, <71, 1>, <89, 1>, <296819808457, 1> ]
 > Factorization(13512748288741441561638433238987525423661915754607935488);
[ <2, 59>, <3, 14>, <7, 6>, <11, 4>, <19, 2>, <23, 1>, <31, 1>, <61, 1>, <181213617120197, 1> ]

Sean.

On 08/31/12 16:43, Paul D Hanna wrote:
> Doug,
>       Fantastic.   I've confirmed all your values up to 940+ digit accuracy.
> I'll submit these terms as 2 sequences after some formula hunting
> and credit you for the computations.
> Thanks!
>      Paul
>
> ---------- Original Message ----------
> From: "D. S. McNeil"<dsm054 at gmail.com>
> To: Sequence Fanatics Discussion list<seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Some Remarkable Series for A193543(n), A193544(n)
> Date: Fri, 31 Aug 2012 00:20:47 -0400
>
>> Can someone find more terms of b(n) and c(n)?
>
> Looks like:
>
> 0 1
> 1 4*sqrt(2) + 4
> 2 96*sqrt(2) + 144
> 3 6912*sqrt(2) + 9792
> 4 903168*sqrt(2) + 1274112
> 5 188891136*sqrt(2) + 267273216
> 6 58042220544*sqrt(2) + 82081714176
> 7 24583741636608*sqrt(2) + 34766166048768
> 8 13730637731069952*sqrt(2) + 19418143011176448
> 9 9777972008057831424*sqrt(2) + 13828133313912766464
> 10 8647047131490907324416*sqrt(2) + 12228770640505737314304
> 11 9297053372804739285123072*sqrt(2) + 13148019425356208796598272
> 12 11943179316739549356493897728*sqrt(2) + 16890206065140879101779771392
> 13 18066284723187322897316620271616*sqrt(2) + 25549584879017650695869778886656
> 14 31785290722166559877155965343105024*sqrt(2) +
> 44951189233408851976112229807292416
> 15 64354805619278535283379331651808002048*sqrt(2) +
> 91011438905480899691423554088718041088
> 16 148568998976062647576057110045620848033792*sqrt(2) +
> 210108293301118675924135051068546238906368
> 17 387932885047590464537298997011364686273183744*sqrt(2) +
> 548619947325464115417997303889817577847783424
> 18 1137529540229602302657361974173461626864204251136*sqrt(2) +
> 1608709703391986198862095660055287572797969137664
> 19 3722061514312152863902279948170443229080820784300032*sqrt(2) +
> 5263789873527529111801402355636596520955646052401152
> 20 13512748288741441561638433238987525423661915754607935488*sqrt(2) +
> 19109911894872035157002147075069564951191053217428406272
>
> to me, but independent verification would be a good idea..
>
>
> Doug
>
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