A094367 -id:A094367 - cp's OEIS frontend

A094366 a(n) is the number of two-generated numerical semigroups whose Frobenius number is 2n-1.

1, 1, 2, 2, 1, 3, 2, 1, 3, 3, 1, 4, 2, 2, 4, 3, 1, 3, 2, 2, 4, 3, 1, 5, 3, 2, 4, 3, 1, 6, 2, 2, 4, 3, 2, 6, 2, 1, 3, 5, 1, 6, 2, 2, 6, 3, 1, 5, 3, 2, 4, 4, 1, 6, 4, 3, 4, 2, 1, 7, 2, 2, 5, 4, 2, 6, 2, 1, 4, 6, 1, 7, 2, 2, 6, 4, 2, 5, 2, 3, 4, 3, 1, 8, 4, 2, 4, 4, 1, 9, 4, 2, 4, 3, 2, 7, 2, 2, 6, 6, 1, 5, 2, 3, 7

Offset: 1

Corina Flynn (Corinamachina(AT)hotmail.com), May 07 2004

A numerical semigroup is a set of natural numbers closed under addition. Its Frobenius number is the largest number not in it. In the case of a semigroup generated by two relatively prime numbers a and b, its Frobenius number is ab-a-b, which is always odd.

			a(9) = 3: the 3 semigroups generated by {2, 19}, {3, 10} and {4, 7} have Frobenius number 17.

David Wasserman, Table of n, a(n) for n = 1..300
J. C. Rosales, P. A. Garcia-Sanchez and J. I. Garcia-Garcia, Every positive integer is the Frobenius number of a numerical semigroup with three generators, Math. Scand. 94 (2004), no. 1, 5-12.

Cf. A094365, A094367.

Edited and extended by David Wasserman, Sep 27 2006

A094365 Number of numerical semigroups with three nonextraneous generators and Frobenius number n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 2, 3, 4, 4, 1, 9, 7, 4, 7, 11, 5, 14, 6, 8, 16, 17, 2, 17, 15, 17, 10, 24, 6, 29, 12, 29, 23, 24, 5, 46, 29, 26, 12, 42, 11, 53, 19, 34, 40, 53, 10, 55, 24, 42, 30, 72, 16, 46, 23, 55, 46, 70, 7, 96, 46, 51, 34, 63, 21, 108, 43, 80, 40, 88, 11, 117, 49, 60

Offset: 1

Kaye A. Archer (godchaser_2(AT)hotmail.com), May 06 2004

nonn

A numerical semigroup is a set of natural numbers closed under addition. Its Frobenius number is the largest number not in it.

A generator is extraneous if it can be generated by other generators.

			a(7)=3 because are three such semigroups with Frobenius number 7. Their complements (and a generating triple) are {1,2,3,7} (4,5,6); {1,2,4,5,7} (3,8,10); {1,2,3,6,7} (4,5,11).

J. L. Davison, On the linear Diophantine problem of Frobenius numbers, J. Number Theory 48 (1994), p353-363.
J. C. Rosales, P. A. Garcia-Sanchez and J. I. Garcia-Garcia, Every positive integer is the Frobenius number of a numerical semigroup with three generators, Math. Scand. 94 (2004), no. 1, 5-12.
J. C. Rosales and M. B. Branco, Irreducible numerical semigroups, Pacific J. Math. 209 (2003), no. 1, 131-143.

Cf. A094366 (2 generators), A094367 (3 generators).

Edited by Don Reble, Apr 26 2007

cp's OEIS Frontend

A094366 a(n) is the number of two-generated numerical semigroups whose Frobenius number is 2n-1.

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