A291095 Smallest k > 1 such that the initial n digits in the decimal expansions of Pi and Pi^k coincide.

3, 3, 878, 11404, 11404, 595413, 1797640, 98274734, 198347106, 8128636028, 75041122922, 922797637351, 6759747953135, 28036830572808, 1213341301344107, 19027704941439533, 71928417857731452, 240751079727999028, 5127701092145711019, 81320964235147379208, 1224942164619356399124

Offset

1,1

Comments

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

Links

Joseph Myers, Table of n, a(n) for n = 1..1000

Chris Thompson, in reply to F. Fröhlich, Re: Just a quick (but hard?) funny sequence idea, SeqFan list, Aug. 2017

Example

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.

Prog

(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

Crossrefs

Cf. A000796.

Sequence in context: A221947 A138662 A009011 * A009498 A076253 A081174

Adjacent sequences: A291092 A291093 A291094 * A291096 A291097 A291098

Keyword

nonn,base

Author

Felix Fröhlich, Aug 30 2017

Extensions

a(3)-a(6) provided by Harvey P. Dale via SeqFan on Aug 26 2017

a(7)-a(8) provided by Daniel Suteu via SeqFan on Aug 26 2017

a(9)-a(10) from Hans Havermann, Aug 31 2017

a(11) onwards from Joseph Myers, Sep 01 2017

Status

approved