[seqfan] Re: Just a quick (but hard?) funny sequence idea

[Felix Fröhlich]

The sequence is now submitted as A291095.

Best regards
Felix

2017-08-30 17:59 GMT+02:00 Chris Thompson <cet1 at cam.ac.uk>:

> On Aug 27 2017, Gaurav Verma wrote:
>
> Here are the first 10 terms of the sequence:
>>
>> a(1) = 3
>> a(2) = 3
>> a(3) = 878
>> a(4) = 11404
>> a(5) = 11404
>> a(6) = 595413
>> a(7) = 1797640
>> a(8) = 98274734
>> a(9) = 298419478
>> a(10) = work in progress, will update soon
>>
>
> The value for a(9) seems to be wrong: pi^298419478 = 31415926 95426383...
> matches only 8 digits of pi.
>
> It's easy to see that each term of the sequence is of the form q+1 where
> p/q is a rather good approximation of log_10(pi). So good that is has to
> be a convergent? A proof of that eludes me at the moment, but certainly
> the values a(1) to a(8) correspond to convergents.
>
> Assuming that conjecture, it is straightforward to compute more terms
> using the continued fraction for log_10(pi) = [0,2,87,4,1,1,1,4,52,...]
>
>   a(1) to a(8) as above
>   a(9) = 198347106
>  a(10) = 8128636028
>  a(11) = 75041122922
>  a(12) = 922797637351
>  a(13) = 11598859508648
>  a(14) = 28036830572808
>  a(15) = 1213341301344107
>  a(16) = 21996765548122104
>  a(17) = 71928417857731452
>  a(18) = 240751079727999028
>  a(19) = 5127701092145711019
>  a(20) = 81320964235147379208
>  a(21) = 1224942164619356399124
>  a(22) = 7268332023480991015532
>  a(23) = 26242236697890514923907
>  a(24) = 1042421135892139605940710
>  a(25) = a(24)
>  a(26) = 44876593316757784085298300
>  a(27) = 1837855483715284868285348842
>  a(28) = 4146393986688580399663359468
>  a(29) = 14747720463039036730368089028
>
> [Computations done with bc(1) using 100 decimal digits.]
>
> If the conjecture is wrong, at least these values are upper limits for
> each a(n).
>
> --
> Chris Thompson
> Email: cet1 at cam.ac.uk
>
>
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