cp's OEIS Frontend

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.

%I A259762 #8 Jul 06 2015 23:39:23
%S A259762 1,2,5,13,29,68,145,307,636,1312,2659,5404,10892,21937,44039,88416,
%T A259762 177136,354965,710576,1422447,2846284,5695248,11393091,22791749,
%U A259762 45588844,91188435,182387991,364797722,729617037,1459278556,2918600648,5837288849,11674666710,23349509456,46699194308,93398744563
%N 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.
%C A259762 In other words, a(n) is the smallest k_1 such that the pairwise products of the n integers satisfy
%C A259762    k_1 * k_1 > k_1 * k_2 > k_1 * k_3 > ... > k_1 * k_n
%C A259762              > k_2 * k_2 > k_2 * k_3 > ... > k_2 * k_n
%C A259762                          > k_3 * k_3 > ... > k_3 * k_n
%C A259762                                                 ...
%C A259762                                            > k_n * k_n.
%C A259762 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).
%F A259762 It appears that lim_{n->inf} a(n)/2^(n-1) = 1.
%e A259762 The positive integer triple (k_1,k_2,k_3) = (5,2,1) yields pairwise products in the required ordering; i.e.,
%e A259762   k_1 * k_1 > k_1 * k_2 > k_1 * k_3
%e A259762             > k_2 * k_2 > k_2 * k_3
%e A259762                         > k_3 * k_3
%e A259762 becomes
%e A259762   5*5 > 5*2 > 5*1
%e A259762       > 2*2 > 2*1
%e A259762             > 1*1
%e A259762 i.e.,
%e A259762   25 > 10 > 5
%e A259762      >  4 > 2
%e A259762           > 1
%e A259762 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.
%e A259762 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.
%e A259762 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.
%Y A259762 Cf. A237749.
%K A259762 nonn
%O A259762 1,2
%A A259762 _Jon E. Schoenfield_, Jul 04 2015