A180026 - OEIS

%I #21 Jan 04 2019 04:57:34

%S 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,

%T 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,

%U 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

%N 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.

%C 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?

%C 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

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1105.3154">Combinatorial minors of matrix functions and their applications</a>, arXiv:1105.3154 [math.CO], 2011-2014.

%H Vladimir Shevelev, <a href="https://www.math.bgu.ac.il/~shevelev/comb_meth2014.pdf">Combinatorial minors of matrix functions and their applications</a>, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).

%F 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).

%e 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.

%Y Cf. A179926, A000005, A001221, A003042.

%K nonn

%O 1,6

%A _Vladimir Shevelev_, Aug 07 2010

%E Corrected and extended by _Alois P. Heinz_ from a(48) via Seqfan Discussion List (Aug 07 2010)