

A066620


Number of unordered triples of distinct pairwise coprime divisors of n.


7



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, 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, 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
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OFFSET

1,12


COMMENTS

a(m) = a(n) if m and n have same factorization structure.


REFERENCES

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 123, Spring 2001.pp 303306.


LINKS



FORMULA

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.


EXAMPLE

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).
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).


MATHEMATICA

Table[Length[Select[Subsets[Divisors[n], {3}], CoprimeQ@@#&]], {n, 100}] (* Gus Wiseman, Apr 28 2021 *)


PROG

(Python)
from sympy import divisor_count as d
def a(n): return (d(n**3)  3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017


CROSSREFS

The version for subsets of {1..n} instead of divisors is A015617.
The nonstrict ordered version is A048785.
The version for pairs of divisors is A063647.
The nonstrict version (3multisets) is A100565.
A version for sets of divisors of any size is A225520.
A007304 ranks 3part strict partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.


KEYWORD

nonn


AUTHOR

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001


EXTENSIONS

Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))


STATUS

approved



