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A351167 Partial sums of A350682.

%I A351167 #36 Feb 17 2024 11:13:26
%S A351167 1,0,0,-1,-1,-1,-2,-2,-2,-3,-3,-3,-4,-4,-3,-4,-4,-4,-4,-3,-3,-4,-4,-3,
%T A351167 -4,-4,-4,-5,-5,-5,-6,-6,-6,-7,-7,-7,-8,-8,-7,-7,-7,-7,-8,-6,-6,-7,-7,
%U A351167 -6,-7,-7,-7,-8,-8,-7,-5,-4,-4,-5,-4,-3,-4,-4,-3,-3,-2,-2,-3,-3,-3,-4,-4,-4,-5,-5,-4,-5,-4,-4,-4,-4,-4,-5,-5,-4,-5,-5,-5,-6,-6,-5,-5,-5,-5,-6,-6,-6,-7,-7,-7,-7
%N A351167 Partial sums of A350682.
%C A351167 Partial sums of Möbius values of triangular numbers under divisibility relation.
%H A351167 Rohan Pandey and Harry Richman, <a href="https://arxiv.org/abs/2402.07934">The Möbius function of the poset of triangular numbers under divisibility</a>, arXiv:2402.07934 [math.NT], 2024. See pp. 2, 8.
%t A351167 Accumulate@ With[{m = 100}, LinearSolve[Table[If[Mod[i (i + 1), j (j + 1)] == 0, 1, 0], {i, m}, {j, m}], UnitVector[m, 1]]] (* _Michael De Vlieger_, Feb 04 2022, after _Harry Richman_ at A350682 *)
%o A351167 (Python)
%o A351167 from sympy import *
%o A351167 triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])
%o A351167 def Mobius_Matrix(lst):
%o A351167     zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]
%o A351167     return Matrix(zeta_array) ** -1
%o A351167 M = Mobius_Matrix(triangular_numbers)
%o A351167 N = M[0, :].tolist()
%o A351167 def sum_function(lst):
%o A351167     sum_list = [sum(lst[:i+1]) for i in range(len(lst))]
%o A351167     return sum_list
%o A351167 S = sum_function(N[0])
%o A351167 print(S)
%o A351167 (PARI) lista(nn) = {my(v=vector(nn, k, k*(k+1)/2)); my(m=matrix(nn, nn, n, k, ! (v[n] % v[k]))); m = 1/m; my(w = vector(nn, k, m[k, 1])); vector(nn-1, k, sum(i=1, k, w[i]));} \\ _Michel Marcus_, Feb 16 2022
%Y A351167 Cf. A000217, A002321, A350682.
%K A351167 sign
%O A351167 1,7
%A A351167 _Rohan Pandey_, _Harry Richman_, Feb 03 2022