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A124506
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Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.
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39
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1, 1, 2, 2, 5, 4, 11, 10, 21, 22, 51, 40, 106, 103, 200, 205, 465, 405, 961, 900, 1828, 1913, 4096, 3578, 8273, 8175, 16132, 16267, 34903, 31822, 70854, 68681, 137391, 140661, 292081, 270258, 591443, 582453, 1156012
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OFFSET
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1,3
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COMMENTS
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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)
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LINKS
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EXAMPLE
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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.
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PROG
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(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.
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CROSSREFS
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Cf. A085489, A088809, A093971, A103580, A116861, A151897, A237668, A308546, A326020, A326083, A364349, A365069.
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KEYWORD
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nonn,more
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AUTHOR
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P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006
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STATUS
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approved
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