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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).

%I A374412 #18 Feb 19 2025 12:06:35
%S A374412 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 A374412 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 A374412 3,4,5,6,7,8,9,10,11,12,13,14,15,16,1,2
%N 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).
%C A374412 Denominators are A204455(n) for row n.
%C A374412 Conjecture 1: The j-th root of the n-th polynomial is:
%C A374412   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 A374412 P(n,x) = denominator(Sum_{h=0..infinity} Sum_{k=1..n} A023900(GCD(n,k))*x^(n*h + k)).
%F A374412 a(n,j) = numerator of exponent of j-th root of [x^m] P(n,x), n >= 0, 0 <= m <= abs(A023900(n)).
%F A374412 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 A374412 The first few polynomial roots are:
%e A374412 {
%e A374412   {1},
%e A374412   {-1},
%e A374412   {-(-1)^(1/3), (-1)^(2/3)},
%e A374412   {-1},
%e A374412   {-(-1)^(1/5), (-1)^(2/5), -(-1)^(3/5), (-1)^(4/5)},
%e A374412   {(-1)^(1/3), -(-1)^(2/3)},
%e A374412   {-(-1)^(1/7), (-1)^(2/7), -(-1)^(3/7), (-1)^(4/7), -(-1)^(5/7), (-1)^(6/7)},
%e A374412   {-1},
%e A374412   {-(-1)^(1/3), (-1)^(2/3)},
%e A374412   {(-1)^(1/5), -(-1)^(2/5), (-1)^(3/5), -(-1)^(4/5)}
%e A374412 }
%e A374412 The irregular triangle a(n,j) begins:
%e A374412   n\j 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 ...
%e A374412   1:  1
%e A374412   2:  1
%e A374412   3:  1  2
%e A374412   4:  1
%e A374412   5:  1  2  3  4
%e A374412   6:  1  2
%e A374412   7:  1  2  3  4  5  6
%e A374412   8:  1
%e A374412   9:  1  2
%e A374412   10: 1  2  3  4
%e A374412   11: 1  2  3  4  5  6  7  8  9 10
%e A374412   12: 1  2
%e A374412   13: 1  2  3  4  5  6  7  8  9 10 11 12
%e A374412   14: 1  2  3  4  5  6
%e A374412   15: 1  2  4  7  8 11 13 14
%e A374412   16: 1
%e A374412   17: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
%e A374412   18: 1  2
%e A374412   ...
%t A374412 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 A374412 (* Conjectured formula: *)
%t A374412   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 A374412 Cf. A374385 (coefficients), A023900, A173557, A204455.
%K A374412 nonn,tabf,frac
%O A374412 1,4
%A A374412 _Mats Granvik_, Jul 08 2024