OFFSET
1,3
COMMENTS
From David W. Wilson, May 17 2017: (Start)
Appears to equal the number of ways to partition Z into n residue classes. For example, a(4) = 10 since we can partition Z into 4 residue classes in 10 ways:
Z = ∪ {0 (mod 2), 1 (mod 4), 3 (mod 8), 7 (mod 8)}
Z = ∪ {0 (mod 2), 3 (mod 4), 1 (mod 8), 5 (mod 8)}
Z = ∪ {0 (mod 2), 1 (mod 6), 3 (mod 6), 5 (mod 6)}
Z = ∪ {1 (mod 2), 0 (mod 4), 2 (mod 8), 6 (mod 8)}
Z = ∪ {1 (mod 2), 2 (mod 4), 0 (mod 8), 4 (mod 8)}
Z = ∪ {1 (mod 2), 0 (mod 6), 2 (mod 6), 4 (mod 6)}
Z = ∪ {0 (mod 3), 1 (mod 3), 2 (mod 6), 5 (mod 6)}
Z = ∪ {0 (mod 3), 2 (mod 3), 1 (mod 6), 4 (mod 6)}
Z = ∪ {1 (mod 3), 2 (mod 3), 0 (mod 6), 3 (mod 6)}
Z = ∪ {0 (mod 4), 1 (mod 4), 2 (mod 4), 3 (mod 4)}
(End)
Unfortunately this conjecture is not correct; it fails at a(13). - Jeffrey Shallit, Nov 19 2017
The correct statement is that a(n) is the number of "natural exact covering systems" of cardinality n. These are covering systems (like the ones in David W. Wilson's comment) that are obtained by starting with the size-1 system x == 0 (mod 1) and successively choosing a congruence and "splitting" it into r >= 2 new congruences. - Jeffrey Shallit, Dec 07 2017
LINKS
David W. Wilson, Table of n, a(n) for n = 1..1000
Yu Hin (Gary) Au, Decompositions of Unit Hypercubes and the Reversion of a Generalized M�bius Series, arXiv:2205.03680 [math.CO], 2022.
I. P. Goulden, Andrew Granville, L. Bruce Richmond, and J. Shallit, Natural exact covering systems and the reversion of the M�bius series, Ramanujan J. (2019) Vol. 50, 211-235.
I. P. Goulden, L. B. Richmond, and J. Shallit, Natural exact covering systems and the reversion of the M�bius series, arXiv:1711.04109 [math.NT], 2017-2018.
N. J. A. Sloane, Transforms
FORMULA
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} mu(k) * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020
MATHEMATICA
InverseSeries[Sum[MoebiusMu[n] x^n, {n, 0, 25}] + O[x]^25] // CoefficientList[#, x]& // Rest (* Jean-Fran�ois Alcover, Sep 29 2018 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Christian G. Bower, Nov 15 1999
STATUS
approved