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

1,12

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. 1-2-3, Spring 2001.pp 303-306.

Links

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

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

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

(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

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.

Sequence in context: A335447 A089233 A343653 * A219023 A025427 A348536

Adjacent sequences: A066617 A066618 A066619 * A066621 A066622 A066623

Keyword

nonn

Author

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

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))

Status

approved