A273525 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.

2, 3, 5, 15, 875, 603919253973

Offset: 0

Author

Vladimir Reshetnikov, May 23 2016

Keywords

• nonn
• more
• hard

Comments

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

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

a(5) = 603919253973 was computed by Japheth Lim (see Math.StackExchange link). - Vladimir Reshetnikov, Aug 23 2016

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.

Examples

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

Links

• Mathematics Stack Exchange, Repeatedly taking mean values of non-empty subsets of a set: 2, 3, 5, 15, 875, ..., May 23 2016.

Programs

Mathematica

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

Extensions

a(5) from Vladimir Reshetnikov, Aug 23 2016