A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

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

Offset: 1

Author

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

Keywords

• nonn

Comments

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

Examples

			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).
		

References

• 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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of zeros are A000961.

Positions of ones are A006881.

The version for subsets of {1..n} instead of divisors is A015617.

The non-strict ordered version is A048785.

The version for pairs of divisors is A063647.

The non-strict version (3-multisets) is A100565.

The version for partitions is A220377 (non-strict: A307719).

A version for sets of divisors of any size is A225520.

A000005 counts divisors.

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.

A018892 counts unordered pairs of coprime divisors (ordered: A048691).

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

A337461 counts 3-part pairwise coprime compositions.

A338331 lists Heinz numbers of pairwise coprime partitions.

Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Programs

Mathematica

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

PARI

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

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

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.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

Extensions

More terms from Vladeta Jovovic, Apr 03 2003

Name corrected by Andrey Zabolotskiy, Dec 09 2020

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