OFFSET
1,3
COMMENTS
From Gus Wiseman, Aug 28 2023: (Start)
Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:
{1} {2} {3} {4} {5} {6} {7} {8}
{2,3} {3,4} {2,5} {4,6} {2,7} {3,8}
{3,5} {5,6} {3,7} {5,8}
{4,5} {4,5,6} {4,7} {6,8}
{3,4,5} {5,7} {7,8}
{6,7} {3,7,8}
{3,5,7} {5,6,8}
{4,5,7} {5,7,8}
{4,6,7} {6,7,8}
{5,6,7} {5,6,7,8}
{4,5,6,7}
Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.
(End)
LINKS
Maria Bras-Amorós and Vicenç Torra, Table of n, a(n) for n = 1..128 (terms 1..100 from Martin Fuller).
Manuel Delgado, Neeraj Kumar, and Claude Marion, On counting numerical semigroups by maximum primitive and Wilf's conjecture, arXiv:2501.04417 [math.CO], 2025. See p. 22.
S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia, and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).
FORMULA
a(n) = Sum_{d|n} B(n/d), where B=1,0,1,1,4,2,10,... is the first differences of A358392. B is A_n in the Delgado link. - Martin Fuller, Dec 18 2025
EXAMPLE
a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.
a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.
a(3) = 2 via <2,5> = {0,2,4,5,...}; and <4,5,6,7> = {0,4,5,6,7,...} where in both the largest missing number is 3.
a(4) = 2 via <3,5,7> = {0,3,5,6,7,...} and <5,6,7,8,9> = {5,6,7,8,9,...} where in both the largest missing number is 4.
PROG
(GAP) The sequence was originally generated by a C program and a Haskell script. The sequence can be obtained by using the function NumericalSemigroupsWithFrobeniusNumber included in the numericalsgps GAP package.
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006
EXTENSIONS
a(40) onward from Martin Fuller, Dec 18 2025
STATUS
approved