OFFSET
1,4
COMMENTS
FORMULA
EXAMPLE
The first few polynomial roots are:
{
{1},
{-1},
{-(-1)^(1/3), (-1)^(2/3)},
{-1},
{-(-1)^(1/5), (-1)^(2/5), -(-1)^(3/5), (-1)^(4/5)},
{(-1)^(1/3), -(-1)^(2/3)},
{-(-1)^(1/7), (-1)^(2/7), -(-1)^(3/7), (-1)^(4/7), -(-1)^(5/7), (-1)^(6/7)},
{-1},
{-(-1)^(1/3), (-1)^(2/3)},
{(-1)^(1/5), -(-1)^(2/5), (-1)^(3/5), -(-1)^(4/5)}
}
The irregular triangle a(n,j) begins:
n\j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
1: 1
2: 1
3: 1 2
4: 1
5: 1 2 3 4
6: 1 2
7: 1 2 3 4 5 6
8: 1
9: 1 2
10: 1 2 3 4
11: 1 2 3 4 5 6 7 8 9 10
12: 1 2
13: 1 2 3 4 5 6 7 8 9 10 11 12
14: 1 2 3 4 5 6
15: 1 2 4 7 8 11 13 14
16: 1
17: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
18: 1 2
MATHEMATICA
nn = 18; f[n_] := DivisorSum[n, MoebiusMu[#] # &]; roots = Table[(x /. Solve[Denominator[Sum[Sum[f[GCD[n, k]]*x^(n*h + k), {k, 1, n}], {h, 0, Infinity}]] == 0, x]), {n, 1, nn}]; Flatten[ReplaceAll[Numerator[Exponent[roots, -1]], 0 -> 1]]
(* Conjectured formula: *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Mats Granvik, Jul 08 2024
STATUS
approved