%I #33 Dec 09 2022 15:39:38

%S 1,4,5,8,11,14,17,22,25,28,33,38,41,46,51,56,61,66,71,76,81,88,93,98,

%T 103,110,117,122,127,134,141,148,153,160,167,174,181,188,195,202,209,

%U 216,223,230,237,244,253,260,267,274,281,290,299

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

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

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

%H Sela Fried, <a href="/A353044/b353044.txt">Table of n, a(n) for n = 1..1000</a>

%H Sela Fried, <a href="https://rdcu.be/c0Wtk">The minimal sum of squares over partitions with a nonnegative rank</a>, Annals of Combinatorics, 2022.

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

%e 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.

%o (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

%Y Cf. A064174, A047993.

%K nonn

%O 1,2

%A _Sela Fried_, Apr 19 2022