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A351167
Partial sums of A350682.
0
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
1,7
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.
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 *)
PROG
(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)
(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
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved