A374412 - OEIS

%I #12 Jul 22 2024 15:20:13

%S 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,

%T 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,

%U 3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2

%N 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).

%C Denominators are A204455(n) for row n.

%C Conjecture 1: The j-th root of the n-th polynomial is:

%C   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.

%F P(n,x) = denominator(Sum_{h=0..infinity} Sum_{k=1..n} A023900(GCD(n,k))*x^(n*h + k)).

%F a(n,j) = numerator of exponent of j-th root of [x^m] P(n,x), n >= 0, 0 <= m <= abs(A023900(n)).

%F Conjecture 1: a(n,j) = j*[GCD(A204455(n), j) = 1], 1 <= j <= A204455(n), where zeros are deleted. Verified up to n = 200.

%e The first few polynomial roots are:

%e {

%e {1},

%e {-1},

%e {-(-1)^(1/3), (-1)^(2/3)},

%e {-1},

%e {-(-1)^(1/5), (-1)^(2/5), -(-1)^(3/5), (-1)^(4/5)},

%e {(-1)^(1/3), -(-1)^(2/3)},

%e {-(-1)^(1/7), (-1)^(2/7), -(-1)^(3/7), (-1)^(4/7), -(-1)^(5/7), (-1)^(6/7)},

%e {-1},

%e {-(-1)^(1/3), (-1)^(2/3)},

%e {(-1)^(1/5), -(-1)^(2/5), (-1)^(3/5), -(-1)^(4/5)}

%e }

%e The irregular triangle a(n,j) begins:

%e n\j 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...

%e 1:  1

%e 2:  1

%e 3:  1  2

%e 4:  1

%e 5:  1  2  3  4

%e 6:  1  2

%e 7:  1  2  3  4  5  6

%e 8:  1

%e 9:  1  2

%e 10: 1  2  3  4

%e 11: 1  2  3  4  5  6  7  8  9 10

%e 12: 1  2

%e 13: 1  2  3  4  5  6  7  8  9 10 11 12

%e 14: 1  2  3  4  5  6

%e 15: 1  2  4  7  8 11 13 14

%e 16: 1

%e 17: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

%e 18: 1  2

%t 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]]

%t (* Conjectured formula: *)

%t   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}]]

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

%K nonn,frac

%O 1,4

%A _Mats Granvik_, Jul 08 2024