A078107 Numbers k such that it is not possible to arrange the numbers from 1 to k in a chain with adjacent links summing to a square.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 18, 19, 20, 21, 22, 24

Offset: 1

Author

R. K. Guy, Dec 06 2002

Keywords

• nonn
• fini
• full

Comments

It seems certain, on account of the valences of the underlying graph, that necklaces exist for all larger k, but this may not yet have been proved.

The problem originated (for k = 15) with Bernardo Recamán Santos of Colombia. The problem for necklaces is due to Joe Kisenwether.

Ed Pegg Jr and W. Edwin Clark have found necklaces (and hence chains) for k = 32 onwards up to 50 and for several larger numbers.

It has been proven that there are no more terms. See A090461 for details. - Paolo Xausa, May 29 2024

Examples

			E.g., for 15, 16 or 17, use (16-)9-7-2-14-11-5-4-12-13-3-6-10-15-1-8(-17).
		

Crossrefs

Cf. A071983, A071984, A090460, A090461.