A350682 Möbius values of triangular numbers under divisibility relation.

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

Author

Rohan Pandey, Harry Richman, Jan 11 2022

Keywords

• sign

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

• Michael De Vlieger, Table of n, a(n) for n = 1..10000.
• 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.
• Wikipedia, Möbius inversion formula on posets.

Crossrefs

Cf. A076982, A008683, A000217.

Programs

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 *)

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

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])