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A083399
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Number of divisors of n that are not divisors of other divisors of n.
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19
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1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4
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OFFSET
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1,2
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COMMENTS
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a(n)<=tau(n); a(n)=tau(n) iff n is prime or n=1 (A008578, A000040); a(n)=tau(n)-1 iff n is semiprime (A001358).
a(n) is the number of maximal subsemigroups of the annular Jones monoid of degree n.
a(n) is the number of maximal subsemigroups of the monoid of orientation-preserving mappings on a set with n elements.
a(n) + 1 is the number of maximal subsemigroups of the monoid of orientation-preserving partial mappings on a set with n elements.
(End)
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LINKS
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FORMULA
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a(n) = omega(n) + 1, where omega = A001221.
G.f.: x/(1 - x) + Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Mar 21 2017
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EXAMPLE
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{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2, 3, 4 and 6 divide not only 24, but also 8 or 12, therefore a(24)=3.
{1,2,3,4,6,8,12,24} are the divisors of n=24: 1, 2 and 3 are noncomposites, therefore a(24)=3. - Jaroslav Krizek, Nov 25 2009
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MAPLE
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1+nops(numtheory[factorset](n)) ;
end proc:
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MATHEMATICA
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Rest@ CoefficientList[ Series[(1/(1 - x)) + Sum[1/(1 - x^Prime[j]), {j, 200}], {x, 0, 111}], x] (* Robert G. Wilson v, Aug 16 2011 *)
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PROG
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(Haskell)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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