

A179926


Number of permutations of the divisors of n of the form d_1=n, d_2, d_3, ..., d_tau(n) such that d_(i+1)/d_i is a prime or 1/prime for all i.


4



1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 18, 1, 1, 2, 2, 2, 8, 1, 2, 2, 4, 1, 18, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 106, 1, 2, 3, 1, 2, 18, 1, 3, 2, 18, 1, 17, 1, 2, 3, 3, 2, 18, 1, 5, 1, 2, 1, 106, 2, 2, 2, 4, 1, 106, 2, 3, 2, 2, 2, 6, 1, 3, 3, 8, 1, 18, 1, 4, 18, 2, 1, 17, 1, 18, 2, 5, 1, 18, 2, 3, 3, 2, 2, 572
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OFFSET

1,6


COMMENTS

In view of formulas given below, there are many common first terms with A001221. Note that, for n>=1, a(n) is positive; it is function of exponents of prime power factorization of n only; moreover, it is invariant with respect to permutations of them.
An equivalent multiset formulation of the problem: for a given finite multiset A, we should, beginning with A, to get all submultisets of A, if, by every step, we remove or join 1 element. How many ways are there to do this?
Via Seqfan Discussion List (Aug 03 2010), Alois P. Heinz proved that every subsequence of the form a(p), a(p*q), a(p*q*r), ..., where p, q, r, ... are distinct primes, coincides with A003043.  Vladimir Shevelev, Aug 09 2010


LINKS

Table of n, a(n) for n=1..120.
V. Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 20112014.
V. Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 516. (2014).


FORMULA

a(p^k)=1, a(p^k*q)=k+1, a(p^2*q^2)=8, a(p^2*q^3)=17, a(pqr)=18, a(p^2*q*r)=106, a(p^3*q*r)=572, etc. (here p,q,r are distinct primes, k>=0).


EXAMPLE

a(12)=3:
[12, 6, 3, 1, 2, 4]
[12, 4, 2, 6, 3, 1]
[12, 4, 2, 1, 3, 6]
a(45)=3:
[45, 15, 5, 1, 3, 9]
[45, 9, 3, 15, 5, 1]
[45, 9, 3, 1, 5, 15]


CROSSREFS

Cf. A000005, A001221, A180026, A003043, A003042.  Vladimir Shevelev, Aug 09 2010
See A173675 for another version.
Sequence in context: A008480 A168324 A285572 * A066882 A068347 A284556
Adjacent sequences: A179923 A179924 A179925 * A179927 A179928 A179929


KEYWORD

nonn


AUTHOR

Vladimir Shevelev, Aug 02 2010


EXTENSIONS

Corrected by D. S. McNeil and Alois P. Heinz and extended by Alois P. Heinz from a(46) via the Seqfan Discussion List (Aug 02 2010)


STATUS

approved



