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Number of maximal pairwise coprime sets of divisors of n.
13

%I #16 Apr 28 2021 07:41:16

%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2,

%T 2,8,1,2,2,6,1,5,1,4,4,2,1,8,2,4,2,4,1,6,2,6,2,2,1,10,1,2,4,6,2,5,1,4,

%U 2,5,1,12,1,2,4,4,2,5,1,8,4,2,1,10,2,2

%N Number of maximal pairwise coprime sets of divisors of n.

%C Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:

%C {6} {30} {6} {30} {30}

%C {12} {2,15} {12} {60} {60}

%C {2,3} {3,10} {18} {2,15} {120}

%C {3,4} {5,6} {36} {3,10} {2,15}

%C {2,3,5} {2,3} {3,20} {3,10}

%C {2,9} {4,15} {3,20}

%C {3,4} {5,6} {3,40}

%C {4,9} {5,12} {4,15}

%C {2,3,5} {5,6}

%C {3,4,5} {5,12}

%C {5,24}

%C {8,15}

%C {2,3,5}

%C {3,4,5}

%C {3,5,8}

%F a(n) = A343660(n) + A005361(n).

%e The a(n) sets for n = 12, 30, 36, 60, 120:

%e {1,6} {1,30} {1,6} {1,30} {1,30}

%e {1,12} {1,2,15} {1,12} {1,60} {1,60}

%e {1,2,3} {1,3,10} {1,18} {1,2,15} {1,120}

%e {1,3,4} {1,5,6} {1,36} {1,3,10} {1,2,15}

%e {1,2,3,5} {1,2,3} {1,3,20} {1,3,10}

%e {1,2,9} {1,4,15} {1,3,20}

%e {1,3,4} {1,5,6} {1,3,40}

%e {1,4,9} {1,5,12} {1,4,15}

%e {1,2,3,5} {1,5,6}

%e {1,3,4,5} {1,5,12}

%e {1,5,24}

%e {1,8,15}

%e {1,2,3,5}

%e {1,3,4,5}

%e {1,3,5,8}

%t fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];

%t Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

%Y The case of pairs is A063647.

%Y The case of triples is A066620.

%Y The non-maximal version counting empty sets and singletons is A225520.

%Y The non-maximal version with no 1's is A343653.

%Y The non-maximal version is A343655.

%Y The version for subsets of {1..n} is A343659.

%Y The case without 1's or singletons is A343660.

%Y A018892 counts pairwise coprime unordered pairs of divisors.

%Y A048691 counts pairwise coprime ordered pairs of divisors.

%Y A048785 counts pairwise coprime ordered triples of divisors.

%Y A084422, A187106, A276187, and A320426 count pairwise coprime sets.

%Y A100565 counts pairwise coprime unordered triples of divisors.

%Y A305713 counts pairwise coprime non-singleton strict partitions.

%Y A324837 counts minimal subsets of {1...n} with least common multiple n.

%Y A325683 counts maximal Golomb rulers.

%Y A326077 counts maximal pairwise indivisible sets.

%Y Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

%K nonn

%O 1,4

%A _Gus Wiseman_, Apr 25 2021