OFFSET
1,30
COMMENTS
Say d is a "floor quotient" of n if d = [n/k] for some positive integer k. This defines a partial order relation on the positive integers. This sequence records the Moebius function values of this poset.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..10000
J.-P. Cardinal, Symmetric matrices related to the Mertens function, arXiv:0811.3701 [math.NT], 2008-2009.
J. C. Lagarias and D. H. Richman, The floor quotient partial order, Adv. Appl. Math., 153 (2024); arXiv:2212.11689 [math.NT], 2022-2023.
EXAMPLE
For n = 9, the set of floor quotients of 9 are Q(9) = {1, 2, 3, 4, 9} with Moebius values a(1) = 1, a(2) = -1, a(3) = -1, and a(4) = 0. The Moebius recursion requires that the Moebius values summed over Q(9) must equal zero, so a(9) = 1.
MATHEMATICA
LinearSolve[Table[If[Floor[i/j] > Floor[i/(j + 1)], 1, 0], {i, n}, {j, n}], UnitVector[n, 1]]
PROG
(PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, my(S=Set(vector(n-1, k, n\(k+1)))); v[n]=-sum(i=1, #S, v[S[i]])); v} \\ Andrew Howroyd, Jan 24 2023
CROSSREFS
KEYWORD
sign,look
AUTHOR
Harry Richman, Jan 24 2023
STATUS
approved