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A100565
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}.
10
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
OFFSET
1,2
COMMENTS
First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
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
LINKS
FORMULA
a(n) = (tau(n^3) + 3*tau(n) + 2)/6.
EXAMPLE
From Gus Wiseman, May 01 2021: (Start)
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)
MATHEMATICA
pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y, {2}]);
Table[Length[Select[Tuples[Divisors[n], 3], LessEqual@@#&&pwcop[#]&]], {n, 30}] (* Gus Wiseman, May 01 2021 *)
PROG
(PARI) A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017
CROSSREFS
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Sequence in context: A337331 A018892 A343654 * A244098 A285573 A325339
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Nov 28 2004
STATUS
approved