|
|
A173675
|
|
Let d_1, d_2, d_3, ..., d_tau(n) be the divisors of n; a(n) = number of permutations p of d_1, d_2, d_3, ..., d_tau(n) such that p_(i+1)/p_i is a prime or 1/prime for i = 1,2,...,tau(n)-1 and p_1 <= p_tau(n).
|
|
4
|
|
|
1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 8, 1, 4, 4, 1, 1, 8, 1, 8, 4, 4, 1, 14, 1, 4, 1, 8, 1, 72, 1, 1, 4, 4, 4, 20, 1, 4, 4, 14, 1, 72, 1, 8, 8, 4, 1, 22, 1, 8, 4, 8, 1, 14, 4, 14, 4, 4, 1, 584, 1, 4, 8, 1, 4, 72, 1, 8, 4, 72, 1, 62, 1, 4, 8, 8, 4, 72, 1, 22, 1, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
Variant of A179926 in which the permutation of the divisors may start with any divisor but the first term may not be larger than the last term.
Equivalently, the number of undirected Hamiltonian paths in a graph with vertices corresponding to the divisors of n and edges connecting divisors that differ by a prime.
a(n) depends only on the prime signature of n. See A295786. (End)
|
|
LINKS
|
|
|
FORMULA
|
a(p^e) = 1 for prime p.
|
|
EXAMPLE
|
a(1) = 1: [1].
a(2) = 1: [1,2].
a(6) = 4: [1,2,6,3], [1,3,6,2], [2,1,3,6], [3,1,2,6].
a(12) = 8: [1,2,4,12,6,3], [1,3,6,2,4,12], [1,3,6,12,4,2], [2,1,3,6,12,4], [3,1,2,4,12,6], [3,1,2,6,12,4], [4,2,1,3,6,12], [6,3,1,2,4,12].
|
|
MAPLE
|
with(numtheory):
q:= (i, j)-> is(i/j, integer) and isprime(i/j):
b:= proc(s, l) option remember; `if`(s={}, 1, add(
`if`(q(l, j) or q(j, l), b(s minus{j}, j), 0), j=s))
end:
a:= proc(n) option remember; ((s-> add(b(s minus {j}, j),
j=s))(divisors(n)))/`if`(n>1, 2, 1)
end:
|
|
MATHEMATICA
|
b[s_, l_] := b[s, l] = If[s == {}, 1, Sum[If[PrimeQ[l/j] || PrimeQ[j/l], b[s ~Complement~ {j}, j], 0], {j, s}]];
a[n_] := a[n] = Function[s, Sum[b[s ~Complement~ {j}, j], {j, s}]][ Divisors[n]] / If[n > 1, 2, 1];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Alois P. Heinz corrected and clarified the definition and provided more terms. - Nov 07 2014
|
|
STATUS
|
approved
|
|
|
|