cp's OEIS Frontend

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

%I A360079 #19 Jun 18 2025 21:02:19
%S A360079 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,
%T A360079 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,
%U A360079 -1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2
%N A360079 Finite differences of Moebius function for the floor quotient poset.
%C A360079 a(n) = mu(n) - mu(n-1), where mu(n) = A360078(n) is the Moebius function of the floor quotient poset.
%H A360079 Andrew Howroyd, <a href="/A360079/b360079.txt">Table of n, a(n) for n = 1..10000</a>
%H A360079 J.-P. Cardinal, <a href="https://arxiv.org/abs/0811.3701">Symmetric matrices related to the Mertens function</a>, arXiv:0811.3701 [math.NT], 2008.
%H A360079 J. C. Lagarias and D. H. Richman, <a href="https://arxiv.org/abs/2212.11689">The floor quotient partial order</a>, arXiv:2212.11689 [math.NT], 2022.
%t A360079 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]]
%o A360079 (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
%o A360079 (Haskell)
%o A360079 isFQ d n = (n `div` (n `div` d)) == d
%o A360079 fqMobius 1 = 1
%o A360079 fqMobius n = - sum [fqMobius d | d <- [1..(n-1)], d `isFQ` n]
%o A360079 a360079 1 = 1
%o A360079 a360079 n = fqMobius n - fqMobius (n-1)
%o A360079 -- _Harry Richman_, Jun 13 2025
%Y A360079 Cf. A002321, A008683, A360078.
%K A360079 sign
%O A360079 1,2
%A A360079 _Harry Richman_, Jan 24 2023