A353044 a(n) is the minimal sum of squares over partitions of n with a nonnegative rank.

1, 4, 5, 8, 11, 14, 17, 22, 25, 28, 33, 38, 41, 46, 51, 56, 61, 66, 71, 76, 81, 88, 93, 98, 103, 110, 117, 122, 127, 134, 141, 148, 153, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 253, 260, 267, 274, 281, 290, 299

Offset: 1

Author

Sela Fried, Apr 19 2022

Keywords

• nonn

Comments

For n not equal to 2, a(n) is the minimal sum of squares over balanced partitions of n.

a(n) is strictly increasing and has parity equal to n.

Examples

			Both (5, 3, 3, 3, 3) and (6, 3, 2, 2, 2, 2) are balanced and have the minimal sum of squares of 61 over balanced partitions of n = 17.
		

Links

• Sela Fried, Table of n, a(n) for n = 1..1000
• Sela Fried, The minimal sum of squares over partitions with a nonnegative rank, Annals of Combinatorics, 2022.

Crossrefs

Cf. A064174, A047993.

Programs

PARI

a(n) = my(m=oo); forpart(p=n, if (vecmax(p) >= #p, m = min(m, norml2(Vec(p))));); m; \\ Michel Marcus, Aug 09 2022

Formula

a(n) = Theta(n^(4/3)).