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A100565
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a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.
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10
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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, 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, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17
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OFFSET
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1,2
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COMMENTS
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Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021
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LINKS
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FORMULA
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a(n) = (tau(n^3) + 3*tau(n) + 2)/6.
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EXAMPLE
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The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
(1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1)
(1,1,2) (1,1,2) (1,1,2) (1,1,2) (1,1,2) (1,1,2)
(1,1,4) (1,1,3) (1,1,4) (1,1,3) (1,1,3)
(1,1,6) (1,1,8) (1,1,4) (1,1,4)
(1,2,3) (1,1,6) (1,1,6)
(1,2,3) (1,1,8)
(1,3,4) (1,2,3)
(1,1,12) (1,3,4)
(1,3,8)
(1,1,12)
(1,1,24)
(End)
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MATHEMATICA
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pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y, {2}]);
Table[Length[Select[Tuples[Divisors[n], 3], LessEqual@@#&&pwcop[#]&]], {n, 30}] (* Gus Wiseman, May 01 2021 *)
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PROG
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CROSSREFS
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The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
A007304 ranks 3-part strict partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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