A273525 - OEIS

%I #29 Aug 20 2022 02:24:28

%S 2,3,5,15,875,603919253973

%N Start with the set {0, 1}. At each step replace the set with the set of means of all its nonempty subsets. a(n) is the size of the set after the n-th step.

%C 2 * 10^6 < a(5) < 7 * 10^12 (see G. Martin's proof at Mathematics Stack Exchange).

%C The brute-force Mathematica program given below overflows for a(5).

%C a(5) = 603919253973 was computed by Japheth Lim (see Math.StackExchange link). -  _Vladimir Reshetnikov_, Aug 23 2016

%C Exactly the same sequence results from the arithmetic mean, geometric mean and harmonic mean, provided that the initial set consists of two distinct positive numbers.

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1797607">Repeatedly taking mean values of non-empty subsets of a set: 2, 3, 5, 15, 875, ...</a>, May 23 2016.

%e Before the first step the set is {0, 1}, so a(0) = 2.

%e After the first step the set is {0, 1, 1/2}, so a(1) = 3.

%e After the second step the set is {0, 1, 1/2, 1/4, 3/4}, so a(2) = 5.

%t Length/@NestList[Union[Mean/@Rest@Subsets@#]&, {0, 1}, 4]

%K nonn,more,hard

%O 0,1

%A _Vladimir Reshetnikov_, May 23 2016

%E a(5) from _Vladimir Reshetnikov_, Aug 23 2016