OFFSET
1,2
COMMENTS
In other words, a(n) is the smallest k_1 such that the pairwise products of the n integers satisfy
k_1 * k_1 > k_1 * k_2 > k_1 * k_3 > ... > k_1 * k_n
> k_2 * k_2 > k_2 * k_3 > ... > k_2 * k_n
> k_3 * k_3 > ... > k_3 * k_n
...
> k_n * k_n.
FORMULA
It appears that lim_{n->inf} a(n)/2^(n-1) = 1.
EXAMPLE
The positive integer triple (k_1,k_2,k_3) = (5,2,1) yields pairwise products in the required ordering; i.e.,
k_1 * k_1 > k_1 * k_2 > k_1 * k_3
> k_2 * k_2 > k_2 * k_3
> k_3 * k_3
becomes
5*5 > 5*2 > 5*1
> 2*2 > 2*1
> 1*1
i.e.,
25 > 10 > 5
> 4 > 2
> 1
which verifies that the requirement is satisfied. The triple (5,3,2) also satisfies the requirement, but there exists no such triple with k_1 < 5, so a(3) = 5.
Similarly, there exist quadruples that meet the requirement (the ones whose largest member is 13 are (13,5,3,2), (13,6,4,3), (13,7,5,4), and (13,8,6,5)), but there is no such quadruple with k_1 < 13, so a(4) = 13.
Of the quintuples that meet the requirement, (29,17,13,11,10) is the only one with k_1 = 29, and there is no such quintuple with k_1 < 29, so a(5) = 29.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jul 04 2015
STATUS
approved