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

Author

Mats Granvik, Jul 08 2024

Keywords

• nonn
• tabf
• frac

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.

Examples

			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
  ...
		

Crossrefs

Cf. A374385 (coefficients), A023900, A173557, A204455.

Programs

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}]]

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.