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A292727 a(n) is the number of states that cannot be achieved when starting from n piles each containing one stone, where stones can be transferred between piles only when they start with the same number of stones.

%I A292727 #28 Feb 22 2023 12:22:44
%S A292727 0,0,1,0,1,3,1,0,3,4,1,7,1,5,9,0,1,14,1,9,17,7,1,26,7,8,30,11,1,55,1,
%T A292727 0,58,10,21,83,1,11,103,30,1,150,1,15,203,13,1,239,15,52,299,17,1,394,
%U A292727 62,34,492,16,1,707,1,17,819,0,107,1021,1,21,1257,187,1,1587
%N A292727 a(n) is the number of states that cannot be achieved when starting from n piles each containing one stone, where stones can be transferred between piles only when they start with the same number of stones.
%C A292727 Note that more than one stone can be moved during a single move.
%C A292727 Conjecture: a(n) = 0 if and only if n is a power of 2.
%C A292727 Conjecture: a(n) = 1 if and only if n is an odd prime.
%H A292727 Bert Dobbelaere, <a href="/A292727/b292727.txt">Table of n, a(n) for n = 1..100</a>
%F A292727 a(n) = A000041(n) - A292726(n).
%F A292727 From _Charlie Neder_, Jan 26 2019: (Start)
%F A292727 a(2^k) = 0.
%F A292727 For p an odd prime, a(p) = 1 and a(2p) = (p+3)/2.
%F A292727 Conjecture: a(4p) = p+4, a(8p) = 2p+20. (End)
%e A292727 For n = 10, the a(10) = 4 partitions of 10 that cannot be generated from transferring stones are: [5, 5], [7, 3], [9, 1], and [10].
%Y A292727 Cf. A000041, A292726.
%K A292727 nonn
%O A292727 1,6
%A A292727 _Peter Kagey_, Sep 21 2017
%E A292727 More terms from _Charlie Neder_, Jan 26 2019
%E A292727 a(61)-a(64) from _Pontus von Brömssen_, Sep 18 2022
%E A292727 More terms from _Bert Dobbelaere_, Feb 22 2023