1,2

Also the number of semigroups of "genus" n.

From Don Zagier's email of Apr 11 1994: (Start)

Given n, one knows that the field of symmetric functions in n variables

a_1,...,a_n is the field Q(sigma_1,...,sigma_n), where

sigma_i is the i-th elementary symmetric polynomial. Here

one has no choice, because sigma_i=0 for i>n and fewer

than n sigma's would not suffice. But, by Newton's formulas,

the field is also given as Q(s_1,...,s_n) where s_i is the

i-th power sum, and now one can ask whether some other sequence

s_{j_1},...,s_{j_n} (0<j_1<...<j_n) also works. For n=1 the

only possibility is clearly s_1, since Q(s_i) = Q(a^i) does

not coincide with Q(a) for i>1, but for n=2 one has two

possibilities Q(s_1,s_2) or Q(s_1,s_3), since from s_1=a+b

and s_3=a^3+b^3 one can reconstruct s_2 = (s_1^3+2s_3)/3s_1.

Similarly, for n=3 one has the possibilities (123), (124),

(125), and (135) (the formula in the last case is

s_2 = (s_1^5+5s_1^2s_3-6s_5)/5(s_1^3-s_3); one can find the

corresponding formulas in the other cases easily) and for

n=4 there are 7: 1234, 1235, 1236, 1237, 1245, 1247, and 1357.

A theorem of Kakutani (I do not know a reference) says that the sequences

which occur are exactly the finite subsets of N whose complements

are additive semigroups (for instance, the complement of {1,2,4,7}

is 3,5,6,8,9,..., which is an closed under addition). This is

a really beautiful theorem. I wrote a simple program to count the sets

of cardinality n which have the property in question for n = 1, ..., 16. (End)

Occurs in Blanco, Justo Puerto, p.6, with a(0) = 1 prepended, as Table 1: "Number of numerical semigroups with given gender" where by "gender" they may mean "Genus." [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 11 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Bras-Amoros, Fibonacci-Like Behavior of the Number of Numerical Semigroups of a Given Genus, Semigroup Forum, 76 (2008), 379-384..

M. Bras-Amoros, Bounds on the Number of Numerical Semigroups of a Given Genus, Journal of Pure and Applied Algebra, Elsevier, vol. 213, n. 6, pp. 997-1001, June 2009. ISSN: 0022-4049. arXiv:0802.2175.

M. Bras-Amoros and S. Bulygin: Towards a Better Understanding of the Semigroup Tree. Semigroup Forum, Springer. Accepted. ISSN: 0037-1912. arXiv:0810.1619.

M. Bras-Amoros and A. de Mier, Representation of Numerical Semigroups by Dyck Paths, Semigroup Forum. (Volume, pages, year?) arXiv:math/0612634.

Sergi Elizalde, Improved bounds on the number of numerical semigroups of a given genus, http://arxiv.org/abs/0905.0489 [From Maria Bras-Amoros, Sep 01 2009]

Jiryo Komeda, Non-Weierstrass numerical semigroups. Semigroup Forum 57 (1998), no. 2, 157-185. [From Maria Bras-Amoros, Sep 01 2009]

Maria Bras-Amoros, Table of n, a(n) for n = 1..50

Maria Bras-Amoros, Home Page [Has many of these references]

Victor Blanco, Justo Puerto, Computing the number of numerical semigroups using generating functions, Jan 9, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 11 2009]

S. R. Finch, Monoids of natural numbers

Nivaldo Medeiros, Numerical Semigroups

Index entries for sequences related to semigroups

Comments from Maria Bras-Amoros (maria.bras(AT)gmail.com), Oct 24 2007, corrected Aug 31 2009: Conjectures: A) a(n) >= a(n-1)+a(n-2); B) a(n)/(a(n-1)+a(n-2)) approaches 1; C) a(n)/a(n-1) approaches the golden ratio.

a(1)=1 because the unique numerical semigroup with genus 1 is N \ {1}

Sequence in context: A072641 A135360 A082548 this_sequence A099604 A026790 A054165

Adjacent sequences: A007320 A007321 A007322 this_sequence A007324 A007325 A007326

nonn,nice

Don Zagier (don.zagier(AT)mpim-bonn.mpg.de), Apr 11 1994

The terms from a(17) onwards were contributed (in the context of semigroups) by Maria Bras-Amoros (maria.bras(AT)gmail.com), Oct 24 2007. The computations were done with the help of Jordi Funollet and Josep M. Mondelo.

Entry revised by N. J. A. Sloane, Aug 31 2009 and Sep 02 2009