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A292726 a(n) is the number of states that can be achieved when starting from n piles each containing one stone, where any number of stones can be transferred between piles that start with the same number of stones.

%I A292726 #26 Jan 29 2019 09:27:29
%S A292726 1,2,2,5,6,8,14,22,27,38,55,70,100,130,167,231,296,371,489,618,775,
%T A292726 995,1254,1549,1951,2428,2980,3707,4564,5549,6841,8349,10085,12300,
%U A292726 14862,17894,21636,26004,31082,37308,44582,53024,63260,75160,88931,105545,124753,147034,173510,204174
%N A292726 a(n) is the number of states that can be achieved when starting from n piles each containing one stone, where any number of stones can be transferred between piles that start with the same number of stones.
%C A292726 This sequence is bounded above by A000041.
%C A292726 Conjecture: A000041(n) - a(n) = 0 if and only if n is a power of 2, and
%C A292726 Conjecture: A000041(n) - a(n) = 1 if and only if n is an odd prime.
%C A292726 Order does not matter: a state containing one pile of two stones and one pile of three stones is considered the same as a state containing one pile of three stones and one pile of two stones.
%C A292726 A056219 is the analogous sequence when only one stone can be moved between piles, and A018819 is the analogous sequence when all stones must be moved between piles. - _Peter Kagey_, Sep 24 2017
%C A292726 Consider the inverse operation: either divide an even stack in two, or replace two stacks of equal parity with their averages. The states which have no inverse moves are precisely those with all parts equal and odd. If n is a power of two, its only odd divisor is 1 and so every state can be inverted to the all-ones state, and if n is an odd prime then the only states which cannot be inverted are the all-ones state and [n]. If n has an odd divisor d with 1 < d < n, then unreachable states include the all-d's state and any state obtainable from it by combining d's in pairs, of which there is at least one. Therefore, both conjectures are true. - _Charlie Neder_, Jan 28 2019
%H A292726 Charlie Neder, <a href="/A292726/b292726.txt">Table of n, a(n) for n = 1..60</a>
%H A292726 Reddit User _AUTOMATIC_, <a href="https://www.reddit.com/r/math/comments/71mbhu/"> Questions about this combinatorial puzzle</a>
%e A292726 For n = 5, the a(5) = 6 states are:
%e A292726 (1 1 1 1 1), (2 1 1 1), (2 2 1), (3 1 1), (3 2), and (4 1).
%e A292726 To reach state (3 2) starting from (1 1 1 1 1):
%e A292726 (1 1 1 1 1) -> (2 1 1 1) -> (2 2 1) -> (3 1 1) -> (3 2).
%Y A292726 Cf. A000041, A018819, A056219.
%K A292726 nonn
%O A292726 1,2
%A A292726 _Peter Kagey_, Sep 21 2017
%E A292726 a(44)-a(50) from _Charlie Neder_, Jan 28 2019