A350682 - OEIS

%I #50 Feb 18 2024 10:06:01

%S 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,

%T -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,

%U -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

%N Möbius values of triangular numbers under divisibility relation.

%C 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.

%H Michael De Vlieger, <a href="/A350682/b350682.txt">Table of n, a(n) for n = 1..10000</a>.

%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, 7.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula#On_posets">Möbius inversion formula on posets</a>.

%t ZetaM = Table[If[Mod[i*(i + 1), j*(j + 1)] == 0, 1, 0], {i, 100}, {j, 100}];

%t MobiusM = LinearSolve[ZetaM, UnitVector[100, 1]] (* _Harry Richman_, Jan 23 2022 *)

%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 print(N[0])

%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; vector(nn, k, m[k, 1]);} \\ _Michel Marcus_, Jan 19 2022

%Y Cf. A076982, A008683, A000217.

%K sign

%O 1,44

%A _Rohan Pandey_, _Harry Richman_, Jan 11 2022