A351167 Partial sums of A350682.

1, 0, 0, -1, -1, -1, -2, -2, -2, -3, -3, -3, -4, -4, -3, -4, -4, -4, -4, -3, -3, -4, -4, -3, -4, -4, -4, -5, -5, -5, -6, -6, -6, -7, -7, -7, -8, -8, -7, -7, -7, -7, -8, -6, -6, -7, -7, -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

Offset: 1

Author

Rohan Pandey, Harry Richman, Feb 03 2022

Keywords

• sign

Comments

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

Links

• Rohan Pandey and Harry Richman, The Möbius function of the poset of triangular numbers under divisibility, arXiv:2402.07934 [math.NT], 2024. See pp. 2, 8.

Crossrefs

Cf. A000217, A002321, A350682.

Programs

Mathematica

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 *)

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

Python

from sympy import *
triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])
def Mobius_Matrix(lst):
    zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]
    return Matrix(zeta_array) ** -1
M = Mobius_Matrix(triangular_numbers)
N = M[0, :].tolist()
def sum_function(lst):
    sum_list = [sum(lst[:i+1]) for i in range(len(lst))]
    return sum_list
S = sum_function(N[0])
print(S)