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Number of pairwise coprime sets of divisors > 1 of n.
7

%I #8 Apr 28 2021 07:41:25

%S 1,2,2,3,2,5,2,4,3,5,2,8,2,5,5,5,2,8,2,8,5,5,2,11,3,5,4,8,2,15,2,6,5,

%T 5,5,13,2,5,5,11,2,15,2,8,8,5,2,14,3,8,5,8,2,11,5,11,5,5,2,25,2,5,8,7,

%U 5,15,2,8,5,15,2,18,2,5,8,8,5,15,2,14,5,5

%N Number of pairwise coprime sets of divisors > 1 of n.

%C First differs from A100565 at a(210) = 52, A100565(210) = 51.

%e The a(n) sets for n = 1, 2, 4, 6, 8, 12, 24, 30, 32, 36, 48:

%e {} {} {} {} {} {} {} {} {} {} {}

%e {2} {2} {2} {2} {2} {2} {2} {2} {2} {2}

%e {4} {3} {4} {3} {3} {3} {4} {3} {3}

%e {6} {8} {4} {4} {5} {8} {4} {4}

%e {2,3} {6} {6} {6} {16} {6} {6}

%e {12} {8} {10} {32} {9} {8}

%e {2,3} {12} {15} {12} {12}

%e {3,4} {24} {30} {18} {16}

%e {2,3} {2,3} {36} {24}

%e {3,4} {2,5} {2,3} {48}

%e {3,8} {3,5} {2,9} {2,3}

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

%e {2,15} {4,9} {3,8}

%e {3,10} {3,16}

%e {2,3,5}

%t pwcop[y_]:=And@@(GCD@@#1==1&)/@Subsets[y,{2}];

%t Table[Length[Select[Subsets[Rest[Divisors[n]]],pwcop]],{n,100}]

%Y The version for partitions is A007359.

%Y The version for subsets of {1..n} is A084422.

%Y The case of pairs is A089233.

%Y The version with 1's is A225520.

%Y The maximal case is A343652.

%Y The case without empty sets or singletons is A343653.

%Y The maximal case without singletons is A343660.

%Y A018892 counts pairwise coprime unordered pairs of divisors.

%Y A051026 counts pairwise indivisible subsets of {1..n}.

%Y A100565 counts pairwise coprime unordered triples of divisors.

%Y A187106, A276187, and A320426 count other types of pairwise coprime sets.

%Y A326077 counts maximal pairwise indivisible sets.

%Y Cf. A007360, A051026, A062319, A074206, A087087, A101268, A285572, A305713, A320423, A326675, A337485, A343655.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 26 2021