A360079 Finite differences of Moebius function for the floor quotient poset.

1, -2, 0, 1, 0, 1, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2

Offset: 1

Author

Harry Richman, Jan 24 2023

Keywords

• sign

Comments

a(n) = mu(n) - mu(n-1), where mu(n) = A360078(n) is the Moebius function of the floor quotient 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.
• J. C. Lagarias and D. H. Richman, The floor quotient partial order, arXiv:2212.11689 [math.NT], 2022.

Crossrefs

Cf. A002321, A008683, A360078.

Programs

Haskell

isFQ d n = (n `div` (n `div` d)) == d
fqMobius 1 = 1
fqMobius n = - sum [fqMobius d | d <- [1..(n-1)], d `isFQ` n]
a360079 1 = 1
a360079 n = fqMobius n - fqMobius (n-1)
-- Harry Richman, Jun 13 2025

Mathematica

LinearSolve[Table[If[Floor[i/j] > Floor[i/(j + 1)], 1, 0], {i, n}, {j, n}] . Table[If[i >= j, 1, 0], {i, n}, {j, n}], UnitVector[n, 1]]

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]])); vector(#v, i, v[i]-if(i>1, v[i-1]))} \\ Andrew Howroyd, Jan 24 2023