1,4

Polya showed that a(n)=n if n is not divisible by 2 and not divisible by 3. Schlude and Specker showed the following: If n is even but not divisible by 3 and not divisible by 4, then a(n)=n-1. If n is divisible by 3, then a(n)<n-1. If n is divisible by 4 but not by 8, then a(n)<n-1. There are open questions as well: Does it hold that a(n)>n-3 for every n? Does it hold that a(n)<n-1 if n is divisible by 8?

Comment from Franklin T. Adams-Watters, Feb 06 2006: According to the Cairns reference, it appears that the questions above were solved by Monsky.

P. Monsky, Problem E3162, Amer. Math. Monthly 96 (1989), 258-259.

G. Polya: Ueber die 'Doppelt-Periodischen' Loesungen des n-Damen-Problems, in: W. Ahrens: Mathematische Unterhaltungen und Spiele, Teubner, Leipzig, 1918, 364-374. Reprinted in: G. Polya: Collected Works, Vol. V, 237-247.

Konrad Schlude and Ernst Specker: Zum Problem der Damen auf dem Torus Technical Report 412 Computer Science Department, ETH Zurich, 2003

Grant Cairns, Queens on Non-square Tori, Electronic Journal of Combinatorics, N6, 2001

Konrad Schlude and Ernst Specker, Zum Problem der Damen auf dem Torus, Technical Report 412, Computer Science Department ETH Zurich, 2003.

a(n) = n, if n is not divisible by 2 and not by 3. a(n) = n-1, if n is divisible by 2, but not by 3 and not by 4 a(n) < n-1, if n is divisible by 3

a(5)=5 because 5 is not divisible by 2 and not divisible by 3

a(10)=10-1=9 because 10 is divisible by 2, but not by 3 and not by 4

a(12)=12-2=10 because 12 is divisible by 3

Cf. A051906, A007705.

Sequence in context: A102513 A100116 A107921 this_sequence A023843 A153990 A154811

Adjacent sequences: A085798 A085799 A085800 this_sequence A085802 A085803 A085804

easy,nonn

Konrad Schlude (schlude(AT)inf.ethz.ch), Jul 24 2003