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%I #32 May 02 2021 20:20:35
%S 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,3,0,1,0,2,0,7,0,0,1,1,
%T 1,4,0,1,1,3,0,7,0,2,2,1,0,4,0,2,1,2,0,3,1,3,1,1,0,13,0,1,2,0,1,7,0,2,
%U 1,7,0,6,0,1,2,2,1,7,0,4,0,1,0,13,1,1,1,3,0,13,1,2,1,1,1,5,0,2,2,4,0,7,0
%N Number of unordered triples of distinct pairwise coprime divisors of n.
%C a(m) = a(n) if m and n have same factorization structure.
%D Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.
%H Antti Karttunen, <a href="/A066620/b066620.txt">Table of n, a(n) for n = 1..10000</a>
%F In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.
%F a(n) = (tau(n^3)-3*tau(n)+2)/6. - _Vladeta Jovovic_, Nov 27 2004
%e a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
%e a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).
%t Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* _Gus Wiseman_, Apr 28 2021 *)
%o (PARI) A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - _Antti Karttunen_, May 27 2017
%o (Python)
%o from sympy import divisor_count as d
%o def a(n): return (d(n**3) - 3*d(n) + 2)/6 # _Indranil Ghosh_, May 27 2017
%Y Positions of zeros are A000961.
%Y Positions of ones are A006881.
%Y The version for subsets of {1..n} instead of divisors is A015617.
%Y The non-strict ordered version is A048785.
%Y The version for pairs of divisors is A063647.
%Y The non-strict version (3-multisets) is A100565.
%Y The version for partitions is A220377 (non-strict: A307719).
%Y A version for sets of divisors of any size is A225520.
%Y A000005 counts divisors.
%Y A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
%Y A007304 ranks 3-part strict partitions.
%Y A014311 ranks 3-part compositions.
%Y A014612 ranks 3-part partitions.
%Y A018892 counts unordered pairs of coprime divisors (ordered: A048691).
%Y A051026 counts pairwise indivisible subsets of {1..n}.
%Y A337461 counts 3-part pairwise coprime compositions.
%Y A338331 lists Heinz numbers of pairwise coprime partitions.
%Y Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.
%K nonn
%O 1,12
%A K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and _Amarnath Murthy_, Dec 24 2001
%E More terms from _Vladeta Jovovic_, Apr 03 2003
%E Name corrected by _Andrey Zabolotskiy_, Dec 09 2020
%E Name corrected by _Gus Wiseman_, Apr 28 2021 (ordered version is 6*a(n))
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