A296195 Number of disjoint covering systems of cardinality n.

1, 1, 3, 10, 39, 160, 691, 3081, 14095, 65757, 311695, 1496833, 7267009

Offset: 1

Author

Jeffrey Shallit, Dec 07 2017

Keywords

• nonn
• more

Comments

A disjoint covering system or DCS (also called "exact covering system" or ECS) is a system of n congruences such that every integer belongs to exactly one of the congruences.

This sequence agrees with A050385 on the first 12 terms, but differs at a(13).

Examples

			For n = 3 the a(3) = 3 DCS are
(i) x == 0 (mod 3), x == 1 (mod 3), x == 2 (mod 3)
(ii) x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 4)
(iii) x == 1 (mod 2), x == 0 (mod 4), x == 2 (mod 4)
		

References

• S. Porubsky and J. Schönheim, Covering systems of Paul Erdös: past, present and future, in Paul Erdös and his Mathematics, Vol. I, Bolyai Society Mathematical Studies 11 (2002), 581-627.

Links

• I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the Möbius series, arXiv:1711.04109v3 [math.NT], revision of Dec 12 2017

Crossrefs

Cf. A050385, which counts a subset of the DCS called "natural exact covering systems".