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%I #83 May 10 2022 07:51:24
%S 1,1,3,10,39,160,691,3081,14095,65757,311695,1496833,7266979,35608419,
%T 175875537,874698246,4376646808,22016578909,111282845162,564886771380,
%U 2878498888625,14719219809915,75505990358779,388451973679785
%N Reversion of Moebius function A008683.
%C From _David W. Wilson_, May 17 2017: (Start)
%C 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:
%C Z = ∪ {0 (mod 2), 1 (mod 4), 3 (mod 8), 7 (mod 8)}
%C Z = ∪ {0 (mod 2), 3 (mod 4), 1 (mod 8), 5 (mod 8)}
%C Z = ∪ {0 (mod 2), 1 (mod 6), 3 (mod 6), 5 (mod 6)}
%C Z = ∪ {1 (mod 2), 0 (mod 4), 2 (mod 8), 6 (mod 8)}
%C Z = ∪ {1 (mod 2), 2 (mod 4), 0 (mod 8), 4 (mod 8)}
%C Z = ∪ {1 (mod 2), 0 (mod 6), 2 (mod 6), 4 (mod 6)}
%C Z = ∪ {0 (mod 3), 1 (mod 3), 2 (mod 6), 5 (mod 6)}
%C Z = ∪ {0 (mod 3), 2 (mod 3), 1 (mod 6), 4 (mod 6)}
%C Z = ∪ {1 (mod 3), 2 (mod 3), 0 (mod 6), 3 (mod 6)}
%C Z = ∪ {0 (mod 4), 1 (mod 4), 2 (mod 4), 3 (mod 4)}
%C (End)
%C Unfortunately this conjecture is not correct; it fails at a(13). - _Jeffrey Shallit_, Nov 19 2017
%C 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
%H David W. Wilson, <a href="/A050385/b050385.txt">Table of n, a(n) for n = 1..1000</a>
%H Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/2205.03680">Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series</a>, arXiv:2205.03680 [math.CO], 2022.
%H I. P. Goulden, Andrew Granville, L. Bruce Richmond, and J. Shallit, <a href="https://doi.org/10.1007/s11139-018-0030-y">Natural exact covering systems and the reversion of the Möbius series</a>, Ramanujan J. (2019) Vol. 50, 211-235.
%H I. P. Goulden, L. B. Richmond, and J. Shallit, <a href="https://arxiv.org/abs/1711.04109">Natural exact covering systems and the reversion of the Möbius series</a>, arXiv:1711.04109 [math.NT], 2017-2018.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Res#revert">Index entries for reversions of series</a>
%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} mu(k) * A(x)^k. - _Ilya Gutkovskiy_, Apr 22 2020
%t InverseSeries[Sum[MoebiusMu[n] x^n, {n, 0, 25}] + O[x]^25] // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Sep 29 2018 *)
%Y Cf. A008683, A050386, A294672, A295162.
%K nonn,nice
%O 1,3
%A _Christian G. Bower_, Nov 15 1999
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