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.

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, 0, 58, 10, 21, 83, 1, 11, 103, 30, 1, 150, 1, 15, 203, 13, 1, 239, 15, 52, 299, 17, 1, 394, 62, 34, 492, 16, 1, 707, 1, 17, 819, 0, 107, 1021, 1, 21, 1257, 187, 1, 1587

Offset: 1

Author

Peter Kagey, Sep 21 2017

Keywords

• nonn

Comments

Note that more than one stone can be moved during a single move.

Conjecture: a(n) = 0 if and only if n is a power of 2.

Conjecture: a(n) = 1 if and only if n is an odd prime.

Examples

			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].
		

Links

• Bert Dobbelaere, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000041, A292726.

Formula

a(n) = A000041(n) - A292726(n).

From Charlie Neder, Jan 26 2019: (Start)

a(2^k) = 0.

For p an odd prime, a(p) = 1 and a(2p) = (p+3)/2.

Conjecture: a(4p) = p+4, a(8p) = 2p+20. (End)

Extensions

More terms from Charlie Neder, Jan 26 2019

a(61)-a(64) from Pontus von Brömssen, Sep 18 2022

More terms from Bert Dobbelaere, Feb 22 2023