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

1,2

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.

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.

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A064174, A047993.

Sequence in context: A047376 A260989 A285566 * A293790 A190778 A117573

Adjacent sequences: A353041 A353042 A353043 * A353045 A353046 A353047

Keyword

nonn

Author

Sela Fried, Apr 19 2022

Status

approved