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A374412
Irregular triangle read by rows: Numerators of exponents of j-th root of the polynomial P(n,x) in A374385, and 1 if n is a power of 2, (numerators of exponents of roots in increasing order).
1
1, 1, 1, 2, 1, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 1, 2, 4, 7, 8, 11, 13, 14, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2
OFFSET
1,4
COMMENTS
Denominators are A204455(n) for row n.
Conjecture 1: The j-th root of the n-th polynomial is:
Root(P(n,x) = 0, j) = -(-1)^(j + n)*(-1)^(j/A204455(n))*[GCD(A204455(n),j) = 1], where 1 <= j <= A204455(n) and where terms equal to 0 are deleted. Conjecture 1 has been verified up to n = 200.
FORMULA
P(n,x) = denominator(Sum_{h=0..infinity} Sum_{k=1..n} A023900(GCD(n,k))*x^(n*h + k)).
a(n,j) = numerator of exponent of j-th root of [x^m] P(n,x), n >= 0, 0 <= m <= abs(A023900(n)).
Conjecture 1: a(n,j) = j*[GCD(A204455(n), j) = 1], 1 <= j <= A204455(n), where zeros are deleted. Verified up to n = 200.
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: *)
nn = 18; A204455[n_] := -(1/2)*(-2 + If[Mod[n, 2] == 0, 1, 0])*Sum[EulerPhi[k]*If[Mod[n, k] == 0, 1, 0]*MoebiusMu[k]^2, {k, 1, n}]; Flatten[Table[DeleteCases[Table[j*If[GCD[A204455[n], j] == 1, 1, 0], {j, 1, A204455[n]}], 0], {n, 1, nn}]]
CROSSREFS
Cf. A374385 (coefficients), A023900, A173557, A204455.
Sequence in context: A181572 A287731 A289816 * A363148 A054482 A208234
KEYWORD
nonn,frac
AUTHOR
Mats Granvik, Jul 08 2024
STATUS
approved