

A353044


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


1



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
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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, arXiv:2204.07873 [math.CO], 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,changed


AUTHOR

Sela Fried, Apr 19 2022


STATUS

approved



