A254432 Natural numbers with the maximum number of "feasible" partitions of length m.

1, 2, 3, 4, 7, 16, 18, 19, 22, 43, 46, 124, 367, 1096, 3283, 9844, 29527, 88576, 265723, 797164, 2391487, 7174456, 21523363, 64570084, 193710247, 581130736, 1743392203, 5230176604

Offset: 1

Author

Md. Towhidul Islam, Jan 30 2015

Keywords

• nonn

Comments

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.

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].

For m>=6, there is a unique natural number with the maximum number of "feasible" partitions.

Examples

			Natural numbers with maximum "feasible" partitions are unique for all m except for m=[2,4,5].
For m=1, the number 1 has 1 "feasible" partition.
For m=2, three numbers 2,3 and 4 each has the highest 1 "feasible" partition.
For m=3, the number 7 has the highest 3 "feasible" partitions.
For m=4, four numbers 16,18,19 and 22 each has the highest 12 "feasible" partitions.
For m=5, two numbers 43 and 46 each has 140 "feasible" partitions.
For m=6, the number 124 has the highest 3950 "feasible" partitions.
For m=7, the number 367 has the highest 263707 "feasible" partitions.
For m=8, the number 1096 has the highest 42285095 "feasible" partitions.
		

Links

• Md Towhidul Islam & Md Shahidul Islam, 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, arXiv:1502.07730 [math.CO], 2015.

Crossrefs

Cf. A254296, A254430, A254431, A254433, A254435, A254436, A254437, A254438, A254439, A254440, A254442.

Formula

For the first 11 values, there is no specific formula.

For n>=12, a(n) = (3^(m-7)+5)/2.

Recursively, for n>=13, a(n) = 3*a(n-1)-5.