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Number of irreducible numerical semigroups with Frobenius number n; that is, irreducible numerical semigroups for which the largest integer not belonging to them is n.
4

%I #16 Feb 16 2025 08:33:09

%S 1,1,1,1,2,1,3,2,3,3,6,2,8,6,7,7,15,7,20,11,18,20,36,14,44,35,45,37,

%T 83,36,109,70,101,106,174,77,246,182,227

%N Number of irreducible numerical semigroups with Frobenius number n; that is, irreducible numerical semigroups for which the largest integer not belonging to them is n.

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a>

%H S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009. [Cached copy, with permission of the author]

%H Calvin Leng, Christopher O'Neill, <a href="https://arxiv.org/abs/1809.09915">A sequence of quasipolynomials arising from random numerical semigroups</a>, arXiv:1809.09915 [math.CO], 2018.

%H J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and J. A. Jimenez-Madrid, <a href="https://doi.org/10.1016/j.jpaa.2003.10.024">Fundamental gaps in numerical semigroups</a>, Journal of Pure and Applied Algebra 189 (2004) 301-313.

%H Clayton Cristiano Silva, <a href="http://www.ime.unicamp.br/~ftorres/ENSINO/MONOGRAFIAS/Clayton.pdf">Irreducible Numerical Semigroups</a>, University of Campinas, São Paulo, Brazil (2019).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FrobeniusNumber.html">Frobenius number</a>

%H <a href="/index/Se#semigroups">Index entries for sequences related to semigroups</a>

%e a(5)=2: the 2 irreducible semigroups generated by {3, 4} and {2, 7} have Frobenius number 5.

%Y Cf. A124506.

%K nonn,more,changed

%O 1,5

%A _Steven Finch_, Mar 13 2009