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, 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.

Links

Table of n, a(n) for n=1..81.

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

Adjacent sequences: A374409 A374410 A374411 * A374413 A374414 A374415

Keyword

nonn,frac

Author

Mats Granvik, Jul 08 2024

Status

approved