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A350682
Möbius values of triangular numbers under divisibility relation.
2
1, -1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 2, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 2, 1, 0, -1, 1, 1, -1, 0, 1, 0, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 0, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0
OFFSET
1,44
COMMENTS
Consider the partial order whose elements are the triangular numbers (T(n) (A000217)) and whose order relation is integer divisibility. Then a(n) is the value mu(T(1), T(n)) of the Möbius function of this partial order.
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, 7.
MATHEMATICA
ZetaM = Table[If[Mod[i*(i + 1), j*(j + 1)] == 0, 1, 0], {i, 100}, {j, 100}];
MobiusM = LinearSolve[ZetaM, UnitVector[100, 1]] (* Harry Richman, Jan 23 2022 *)
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()
print(N[0])
(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; vector(nn, k, m[k, 1]); } \\ Michel Marcus, Jan 19 2022
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved