A362829 Positions in lexicographic order of odd partitions of sufficiently large numbers.

1, 3, 7, 10, 15, 20, 27, 30, 39, 41, 51, 56, 69, 72, 75, 93, 95, 101, 123, 128, 132, 134, 160, 163, 166, 172, 176, 212, 214, 220, 227, 229, 273, 278, 282, 284, 291, 297, 353, 356, 359, 365, 369, 379, 382, 384, 453, 455, 461, 468, 470, 481, 483, 490, 579, 584

Offset: 1

Author

Richard Peterson, Aug 01 2023

Keywords

• nonn

Comments

a(n) is the position in lexicographic order of the n-th odd partition of a sufficiently large number k. As long as the number k whose partitions we are examining is large enough, a(n) will exist and won't change for different k. The number of partitions of an odd number, for example, 101 for k=13, will always appear in the sequence, since 13 is the 101st partition in lexicographic order.

Equivalently, positions of partitions with all parts odd among all partitions with no parts of size 1, ordered first by sum, then lexicographically (with the parts in nondecreasing order); or positions of partitions with all parts even among all partitions ordered first by the number of parts plus the sum of the parts, then lexicographically. - Pontus von Brömssen, Sep 14 2023

Examples

			a(1)=1 because 1+1+...+1 (k times) is the first partition in lexicographic order of any positive integer k, and it is odd.
a(2)=3 because 1+1+...+1(k-3 times)+3=k is the third partition of k lexicographically and it is odd.
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000009, A087897, A000041.

Extensions

More terms from Pontus von Brömssen, Sep 14 2023