%I #35 Oct 19 2024 15:57:32

%S 72,108,120,144,168,180,200,216,240,252,264,270,280,288,300,312,324,

%T 336,360,378,392,396,400,408,420,432,440,450,456,468,480,500,504,520,

%U 528,540,552,560,576,588,594,600,612,616,624,630,648,660,672,675,680

%N Numbers k such that the cyclic group of order k is not a Hajós group.

%C Old name was "Orders of non-Hajós groups".

%D F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 94.

%H Charles R Greathouse IV, <a href="/A102562/b102562.txt">Table of n, a(n) for n = 1..10000</a>

%H Emmanuel Amiot, <a href="http://canonsrythmiques.free.fr/pdf/RhythmicCanonsPNM.pdf"> Structures, Algorithms and Algebraic Tools for Rhythmic Canons, Perspectives of New Music, 2012</a>, LAMPS, Perpignan, France (see page 6).

%H Moreno Andreatta, <a href="http://articles.ircam.fr/textes/Andreatta07a/index.pdf">De la conjecture de Minkowski aux canons rythmiques mosaïques</a>, IRCAM, Paris, (see page 6).

%H Christophe Cordero, <a href="https://arxiv.org/abs/2301.13566">Factorizations of Cyclic Groups and Bayonet Codes</a>, arXiv:2301.13566 [math.CO], 2023, p. 4.

%H Jeremy Kastine, <a href="https://doi.org/10.1007/978-3-030-21392-3_25">Maximally Even Tilings</a>, International Conference on Mathematics and Computation in Music (MCM 2019), Lecture Notes in Computer Science, Vol. 11502, Springer, Cham, 309-321.

%H Jeffrey C. Lagarias and Yang Wang, <a href="http://dx.doi.org/10.1006/jfan.1996.3008">Spectral Sets and Factorizations of Finite Abelian Groups</a>, Journal of Functional Analysis, Volume 145, Issue 1, Apr 01 1997, pp. 73-98 (see page 88).

%H Marie Lhuissier, <a href="https://images-archive.math.cnrs.fr/Canons-rythmiques-mosaiques.html">Canons rythmiques mosaïques</a>, Images des Mathématiques, CNRS, 2023.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HajosGroup.html">Hajos Group</a>

%F a(n) = n + O(n(log log n)^3/log n). - _Charles R Greathouse IV_, Mar 24 2014

%o (PARI) is(n)=my(f=vecsort(factor(n)[,2])~); #f>1 && f!=[2,2] && (#f>2 || f[1]>1) && (#f!=3 || f[2]>1 || f[3]>2) && f!=[1,1,1,1] \\ _Charles R Greathouse IV_, Mar 24 2014

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Jan 14 2005

%E New definition by _Charles R Greathouse IV_, Mar 24 2014