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Partial sums of A350682.
0

%I #36 Feb 17 2024 11:13:26

%S 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 -4,-4,-4,-5,-5,-5,-6,-6,-6,-7,-7,-7,-8,-8,-7,-7,-7,-7,-8,-6,-6,-7,-7,

%U -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 Partial sums of A350682.

%C Partial sums of Möbius values of triangular numbers under divisibility relation.

%H 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 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 (Python)

%o from sympy import *

%o triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])

%o def Mobius_Matrix(lst):

%o zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]

%o return Matrix(zeta_array) ** -1

%o M = Mobius_Matrix(triangular_numbers)

%o N = M[0, :].tolist()

%o def sum_function(lst):

%o sum_list = [sum(lst[:i+1]) for i in range(len(lst))]

%o return sum_list

%o S = sum_function(N[0])

%o print(S)

%o (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 Cf. A000217, A002321, A350682.

%K sign

%O 1,7

%A _Rohan Pandey_, _Harry Richman_, Feb 03 2022