A259762 Smallest integer k_1 such that there exist n positive integers k_1 > k_2 > ... > k_n having the property that k_j * k_n > k_(j+1)^2 for j=1..n-1.

1, 2, 5, 13, 29, 68, 145, 307, 636, 1312, 2659, 5404, 10892, 21937, 44039, 88416, 177136, 354965, 710576, 1422447, 2846284, 5695248, 11393091, 22791749, 45588844, 91188435, 182387991, 364797722, 729617037, 1459278556, 2918600648, 5837288849, 11674666710, 23349509456, 46699194308, 93398744563

Offset: 1

Author

Jon E. Schoenfield, Jul 04 2015

Keywords

• nonn

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.

This is one of the orderings of the pairwise products of real numbers in A237749. Conjecture: if we constrain those real numbers to take integer values, then all A237749(n) orderings of pairwise products can be obtained with k_1 = a(n), but this ordering cannot be obtained with k_1 < a(n).

Examples

			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

Cf. A237749.

Formula

It appears that lim_{n->inf} a(n)/2^(n-1) = 1.