|
%I #25 Apr 17 2016 11:51:58
%S 1,2,3,4,7,16,18,19,22,43,46,124,367,1096,3283,9844,29527,88576,
%T 265723,797164,2391487,7174456,21523363,64570084,193710247,581130736,
%U 1743392203,5230176604
%N Natural numbers with the maximum number of "feasible" partitions of length m.
%C Sequence A254296 describes "feasible" partitions and gives the number of all "feasible" partitions of all natural numbers. We must take the value of m from there.
%C Here we list the natural numbers with the highest number of "feasible" partitions of length m. Such numbers are unique for all m except for m=[2,4,5].
%C For m>=6, there is a unique natural number with the maximum number of "feasible" partitions.
%H Md Towhidul Islam & Md Shahidul Islam, <a href="http://arxiv.org/abs/1502.07730">Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance</a>, arXiv:1502.07730 [math.CO], 2015.
%F For the first 11 values, there is no specific formula.
%F For n>=12, a(n) = (3^(m-7)+5)/2.
%F Recursively, for n>=13, a(n) = 3*a(n-1)-5.
%e Natural numbers with maximum "feasible" partitions are unique for all m except for m=[2,4,5].
%e For m=1, the number 1 has 1 "feasible" partition.
%e For m=2, three numbers 2,3 and 4 each has the highest 1 "feasible" partition.
%e For m=3, the number 7 has the highest 3 "feasible" partitions.
%e For m=4, four numbers 16,18,19 and 22 each has the highest 12 "feasible" partitions.
%e For m=5, two numbers 43 and 46 each has 140 "feasible" partitions.
%e For m=6, the number 124 has the highest 3950 "feasible" partitions.
%e For m=7, the number 367 has the highest 263707 "feasible" partitions.
%e For m=8, the number 1096 has the highest 42285095 "feasible" partitions.
%Y Cf. A254296, A254430, A254431, A254433, A254435, A254436, A254437, A254438, A254439, A254440, A254442.
%K nonn
%O 1,2
%A _Md. Towhidul Islam_, Jan 30 2015
|
