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A180026 a(n) is the number of arrangements of all divisors of n of the form d_1=n, d_2, d_3, ..., d_tau(n) such that every ratio d_(i+1)/d_i and d_tau(n)/d_1 is prime or 1/prime. 3
0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 2, 1, 12, 1, 0, 2, 2, 2, 0, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 2, 0, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 44, 1, 2, 2, 0, 2, 12, 1, 2, 2, 12, 1, 4, 1, 2, 2, 2, 2, 12, 1, 2, 0, 2, 1, 44, 2, 2, 2, 2, 1, 44, 2, 2, 2, 2, 2, 2, 1, 2, 2, 0, 1, 12, 1, 2, 12, 2, 1, 4, 1, 12, 2, 2, 1, 12, 2, 2, 2, 2, 2, 164, 0, 2, 2, 2, 0, 44, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

a(n) depends on 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 and such that, joining to the last submultiset one element, we again obtain A. How many ways to do this?

Via Seqfan Discussion List (Aug 07 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 A003042. - Vladimir Shevelev, Nov 07 2014

LINKS

Table of n, a(n) for n=1..128.

V. Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 2011-2014.

V. Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).

FORMULA

a(p)=1, and, for k>=2, a(p^k)=0; a(p*q)=a(p^2*q)=a(p^3*q)=2; a(p^2*q^2)=0; a(p*q*r)=12, etc. (here p,q,r are distinct primes).

EXAMPLE

If n=p*q, then we have exactly two required chains: p*q, p, 1, q and p*q, q, 1, p. Thus a(6)=a(10)=a(14)=...=2.

CROSSREFS

Cf. A179926, A000005, A001221, A003042.

Sequence in context: A177207 A161528 A175083 * A136176 A291914 A281081

Adjacent sequences:  A180023 A180024 A180025 * A180027 A180028 A180029

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Aug 07 2010

EXTENSIONS

Corrected and extended by Alois P. Heinz from a(48) via Seqfan Discussion List (Aug 07 2010)

STATUS

approved

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Last modified September 25 21:33 EDT 2017. Contains 292500 sequences.