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User: Rohan Pandey

Rohan Pandey has authored 2 sequences.

A351167 Partial sums of A350682.

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

Rohan Pandey, Harry Richman, Feb 03 2022

sign

Partial sums of Möbius values of triangular numbers under divisibility relation.

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.

Cf. A000217, A002321, A350682.

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 *)

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

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)

A350682 Möbius values of triangular numbers under divisibility relation.

Original entry on oeis.org

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

Rohan Pandey, Harry Richman, Jan 11 2022

sign

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.

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.

Cf. A076982, A008683, A000217.

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 *)

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

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])

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User: Rohan Pandey

A351167 Partial sums of A350682.

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