Minimizing...

A friend asked me this math question (presumably popped out of QFT or something)

Consider a set of N elements: {a_i}
It is known that sum(a_i) = 0
and that sum(a_i²) = 1
What is the minimum of sum(a_i⁴)?

The AM-RMS/power mean inequality tells us that sqrt(sum(a_i⁴)/N) >= sum(a_i²)/N ->
sum(a_i⁴) >= 1/N, with equality iff the a_is are the same.

If N is even, then the a_is just alternate as +-1/sqrt(N)
If N is odd, then that trick doesn't work, since then the sum won't = 0.  Therein lies the problem.  It seems likely that the answer has (N+1)/2 of the a_is equaling one number, and the remainder equaling something else.  However, I haven't been able to prove that this is the optimal solution yet.
I tried brute forcing this for N=3, but after an hr of algebra... the answer is always 1/2, regardless of how I pick the a_is.

I dunno lol, this is trickier than it looks.  I think Lagrange multipliers are probably the thing to try, but I hate calculus so I cba to work it out atm :P.  Here's what I get numerically in mathematica:


n  | min
--------------------------------------------------------------------
2  | 0.49999999999999999999999998707530292885894258013424 = 1/2
3  | 0.49999999999999999999999998707530292885894258013424 = 1/2
4  | 0.24999999999999999999999998707530292885894258013425 = 1/4
5  | 0.23333333333333333333333332207448610691712331425027 = 7/30
6  | 0.16666666666666666666666666092235685727064114672633 = 1/6
7  | 0.15476190476190476190476189980890293645614806358886 = 13/84
8  | 0.15833333333333333333333332814909373035342030158718 = 19/120 (should be 1/8)
9  | 0.11666666666666666666666665540781944025045664758361 = 7/60
10 | 0.09999999999999999999999999172819387446972325128590 = 1/10
11 | 0.09393939393939393939393938663984785433003126215405 = 31/330
12 | 0.08333333333333333333333332758902352393730781339299 = 1/12
13 | 0.07875457875457875457875457362416994522082621919912 = 43/546
14 | 0.07142857142857142857142856720826218085189961800301 = 1/14
15 | 0.06785714285714285714285713904831376107598226239058 = 19/280
16 | 0.06646825396825396825396825031374621111412151941437 = 67/1008 (should be 1/16)
17 | 0.05964052287581699346405228463942455072362360671260 = 73/1224
18 | 0.05555555555555555555555555300252897360176643558207 = 1/18
19 | 0.05321637426900584795321637192644483685522992523848 = 91/1710
20 | 0.04999999999999999999999999793204846861743081282147 = 1/20


f[p_, n_] := Sum[x[i]^p, {i, n}]
g[n_] := NMinimize[{f[4, n], f[1, n] == 0, f[2, n] == 1}, Array[x, n], WorkingPrecision -> 50][[1]]

The function NMinimize[] is unreliable (the answers for 8 and 16 are wrong at least), but the exact function Minimize[] is too slow to use. I could only double-check up to N=7, but no further.  The rest of the even ones are correct anyway.  It's interesting that they all seem to be rational, though.