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

[Chris Thompson]

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