[seqfan] Re: slowest-growing "accelerating" sequence

[Allan Wechsler]

I am intrigued by Don Reble's comment about the similar sequence beginning
(1, 1, ,,,) growing faster than Wild's sequence starting (1, 2, ...). This
suggests that the "greedy" algorithm of selecting the smallest next number
that satisfies the constraints does not minimize subsequent entries. Is it
possible that starting (1, 3, ...) makes subsequent entries even smaller?
Or perhaps later non-optimal choices can reduce even later entries?


On Fri, Sep 13, 2013 at 8:49 AM, Don Reble <djr at nk.ca> wrote:

> a(n) = 1,2,5,16,68,404,3587,51747,**1343181,70863530...
>>
>> ... a(n) is the slowest growing integer sequence beginning with 1
>>
>> whose sequences of mth-order quotients are all strictly increasing,
>> for all values of m.
>>
>> jon wild
>>
>
>    Perhaps Jon means _all_ values of m, including 0.
>    (So that it can't begin 1,1.)  Or perhaps not,
>        1 1 2  9 103 3405 371146 153037899 275315985020 2505668630734788
>        1 2 5 16  68  404   3587     51747      1343181         70863530
>    because the 1,1 sequence grows faster anyway. Jon?
>
>    BTW, corresponding difference sequences are A000079 and A000225.
>
>  ...take the product of the previous element a(n) and all of the most
>>
>> recently calculated mth ratios, add 1, and take the floor.
>>
>> --Graeme McRae
>>
>
>    Yes. The growth rate is hyperexponential, so my computer runs out of
>    memory pretty quickly. (Those quotients are killers, not the sequence
>    itself.) Have s'more terms.
>
> 1
> 2
> 5
> 16
> 68
> 404
> 3587
> 51747
> 1343181
> 70863530
> 8771738549
> 3010577465956
> 3472890614331263
> 16778636239137208665
> 435866021017987040200927
> 80724349462285706161330870948
> 146347653618405076026597153184**017178
> 370281025373267805703902488419**0121454618096
> 194146390420750116019045958924**0443604643356373706327
> 327264872668416385059974502188**994396154497307865899952930516**57
> 288446581117129009565353747088**521585632897325578890700993393**
> 37027912646286
> 227469456701027991010652297470**211018056216119398577688089952**
> 1251905648932937035326556839
> 290075547022486333813693535531**230950983289112758751771686880**
> 396844514243266765608885248344**78778858475108
> 114654236288391623803120048812**338378923201909025140523249822**
> 058948949003096097601289509973**594977341379753585511691703312**066
> 286771527322364315637265554893**187362324790934083293394580448**
> 968848634100735335827561955127**136999494818157671130578737141**
> 568527819282103471406137
> 991640462838273375627738696091**672948855453024266252499046892**
> 313948814248377868053876458901**350496387620054463160116560544**
> 010813779680789761429009802800**978591899964162597
>
> --
> Don Reble  djr at nk.ca
>
>
>
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