A291095 - OEIS

%I #57 Jan 02 2023 12:30:54

%S 3,3,878,11404,11404,595413,1797640,98274734,198347106,8128636028,

%T 75041122922,922797637351,6759747953135,28036830572808,

%U 1213341301344107,19027704941439533,71928417857731452,240751079727999028,5127701092145711019,81320964235147379208,1224942164619356399124

%N Smallest k > 1 such that the initial n digits in the decimal expansions of Pi and Pi^k coincide.

%C As pointed out by Chris Thompson on the SeqFan List, the terms of the sequence are of the form q+1, where p/q are increasingly better approximations of log_10(Pi), and apparently (for all known terms*) convergents of the continued fraction. At least this easily yields upper bounds for the terms. [*not true for n=13, 16, 27, 39, ...] - _M. F. Hasler_, Aug 31 2017

%H Joseph Myers, <a href="/A291095/b291095.txt">Table of n, a(n) for n = 1..1000</a>

%H Chris Thompson, in reply to F. Fröhlich, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-August/017908.html">Re: Just a quick (but hard?) funny sequence idea</a>, SeqFan list, Aug. 2017

%e For n = 3: The decimal expansion of Pi starts 3.1415926... and the decimal expansion of Pi^878 starts 31447652... Since 878 is the smallest k > 1 such that the initial three digits in the decimal expansions of Pi and Pi^k are equal, a(3) = 878.

%o (PARI) {c=contfrac(LP=log(Pi)/log(10)); n=0; for(i=1, #c=contfracpnqn(c, #c)[2, ], while(floor(10^(frac((c[i]+1)*LP)+n))==Pi\.1^n, n++; print1(c[i]+1, ", ")))} \\ Yields a non-optimal upper bound for n=13, 16, 27, ... \\ _M. F. Hasler_, Aug 31 2017

%Y Cf. A000796.

%K nonn,base

%O 1,1

%A _Felix Fröhlich_, Aug 30 2017

%E a(3)-a(6) provided by _Harvey P. Dale_ via SeqFan on Aug 26 2017

%E a(7)-a(8) provided by _Daniel Suteu_ via SeqFan on Aug 26 2017

%E a(9)-a(10) from _Hans Havermann_, Aug 31 2017

%E a(11) onwards from _Joseph Myers_, Sep 01 2017