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A350682 Möbius values of triangular numbers under divisibility relation.

%I A350682 #50 Feb 18 2024 10:06:01
%S A350682 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 A350682 -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 A350682 -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 A350682 Möbius values of triangular numbers under divisibility relation.
%C A350682 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 A350682 Michael De Vlieger, <a href="/A350682/b350682.txt">Table of n, a(n) for n = 1..10000</a>.
%H A350682 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 A350682 Wikipedia, <a href="https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula#On_posets">Möbius inversion formula on posets</a>.
%t A350682 ZetaM = Table[If[Mod[i*(i + 1), j*(j + 1)] == 0, 1, 0], {i, 100}, {j, 100}];
%t A350682 MobiusM = LinearSolve[ZetaM, UnitVector[100, 1]] (* _Harry Richman_, Jan 23 2022 *)
%o A350682 (Python)
%o A350682 from sympy import *
%o A350682 triangular_numbers = ([(x * (x + 1) // 2) for x in range(1, 101)])
%o A350682 def Mobius_Matrix(lst):
%o A350682     zeta_array = [[0 if n % m != 0 else 1 for n in lst] for m in lst]
%o A350682     return Matrix(zeta_array) ** -1
%o A350682 M = Mobius_Matrix(triangular_numbers)
%o A350682 N = M[0, :].tolist()
%o A350682 print(N[0])
%o 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; vector(nn, k, m[k, 1]);} \\ _Michel Marcus_, Jan 19 2022
%Y A350682 Cf. A076982, A008683, A000217.
%K A350682 sign
%O A350682 1,44
%A A350682 _Rohan Pandey_, _Harry Richman_, Jan 11 2022