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

1,6

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.

Links

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

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)

Example

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].

Crossrefs

Cf. A000041, A292726.

Sequence in context: A336090 A255123 A244483 * A049403 A104556 A116089

Adjacent sequences: A292724 A292725 A292726 * A292728 A292729 A292730

Keyword

nonn

Author

Peter Kagey, Sep 21 2017

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

Status

approved