A050385 Reversion of Moebius function A008683.

1, 1, 3, 10, 39, 160, 691, 3081, 14095, 65757, 311695, 1496833, 7266979, 35608419, 175875537, 874698246, 4376646808, 22016578909, 111282845162, 564886771380, 2878498888625, 14719219809915, 75505990358779, 388451973679785

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 Möbius 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 Möbius 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 Möbius series, arXiv:1711.04109 [math.NT], 2017-2018.

N. J. A. Sloane, Transforms

Index entries for reversions of series

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-François Alcover, Sep 29 2018 *)

Crossrefs

Cf. A008683, A050386, A294672, A295162.

Sequence in context: A151067 A007163 A219263 * A296195 A123768 A005750

Adjacent sequences: A050382 A050383 A050384 * A050386 A050387 A050388

Keyword

nonn,nice

Author

Christian G. Bower, Nov 15 1999

Status

approved