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A332509 a(n) = Sum_{k=1..n} mu(floor(n/k)), where mu = A008683.

%I A332509 #10 Feb 16 2025 08:33:59
%S A332509 1,0,1,1,1,2,1,2,3,2,1,4,4,3,3,3,3,5,5,6,6,3,2,7,7,6,6,6,6,7,8,9,9,7,
%T A332509 5,8,8,8,8,9,9,11,12,11,11,9,8,13,12,11,12,11,11,13,12,12,13,11,10,15,
%U A332509 15,16,15,16,16,14,15,15,15,12,13,19,19,19,18,19,17,16,17,18,18,17,16,21,21
%N A332509 a(n) = Sum_{k=1..n} mu(floor(n/k)), where mu = A008683.
%H A332509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%F A332509 G.f.: (1/(1 - x)) * (x - Sum_{k>=2} mu(k-1) * x^k / (1 - x^k)).
%F A332509 a(n) = 1 - Sum_{k=1..n} Sum_{d|k, d > 1} mu(d-1) for n > 0.
%F A332509 Sum_{k=1..n-1} mu(k) * a(floor(n/k)) = 0.
%t A332509 Table[Sum[MoebiusMu[Floor[n/k]], {k, 1, n}], {n, 1, 85}]
%t A332509 Table[1 - Sum[DivisorSum[k, MoebiusMu[# - 1] &, # > 1 &], {k, 1, n}], {n, 1, 85}]
%t A332509 nmax = 85; CoefficientList[Series[(1/(1 - x)) (x - Sum[MoebiusMu[k - 1] x^k/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x] // Rest
%o A332509 (PARI) a(n) = sum(k=1, n, moebius(n\k)); \\ _Michel Marcus_, Feb 14 2020
%Y A332509 Cf. A002321, A006218, A008683, A098018, A317625, A332510.
%K A332509 nonn
%O A332509 1,6
%A A332509 _Ilya Gutkovskiy_, Feb 14 2020