A294672 Number of disjoint covering systems of cardinality n, up to equivalence under shift.

1, 1, 2, 4, 10, 26, 75, 226, 718, 2368, 8083, 28367

Offset

1,3

Comments

A disjoint covering system is a system of n congruences x == a_i (mod m_i) such that every integer is a solution to exactly one of the congruences. This sequence counts them up to "shift"; that is, two systems are the same if we can turn one into another by subtracting a constant from x.

Links

Table of n, a(n) for n=1..12.

I. P. Goulden, Andrew Granville, L. Bruce Richmond, and Jeffrey Shallit, Natural exact covering systems and the reversion of the Möbius series, Ramanujan J. (2019) Vol. 50, 211-235.

Břetislav Novák and Štefan Znám, Disjoint Covering Systems, The American Mathematical Monthly, Vol. 81, No. 1 (1974), 42-45.

Wikipedia, Covering system.

Example

For n = 3 there are three disjoint covering systems:
(a) x == 0 (mod 3), x == 1 (mod 3), x == 2 (mod 3)
(b) x == 0 (mod 2), x == 1 (mod 4), x == 3 (mod 4)
(c) x == 1 (mod 2), x == 0 (mod 4), x == 2 (mod 4)
but (b) and (c) are equivalent under shift.

Crossrefs

Sequence in context: A006123 A006251 A049401 * A239077 A148099 A007579

Adjacent sequences: A294669 A294670 A294671 * A294673 A294674 A294675

Keyword

nonn,more

Author

Jeffrey Shallit, Nov 06 2017

Status

approved