cp's OEIS Frontend

A332510 a(n) = Sum_{k=1..n} lambda(floor(n/k)), where lambda = A008836.

%I A332510 #9 Feb 16 2025 08:33:59
%S A332510 1,0,1,2,1,2,1,2,5,2,1,4,5,4,3,4,3,6,7,6,7,4,3,8,7,6,7,8,9,8,9,10,11,
%T A332510 8,5,10,9,10,11,10,9,12,13,12,13,12,11,16,17,12,13,12,13,16,13,14,15,
%U A332510 14,13,16,15,16,17,20,19,18,19,18,19,14,15,22,23,22,19,22,21,20,21,20,23,20,19,26,23
%N A332510 a(n) = Sum_{k=1..n} lambda(floor(n/k)), where lambda = A008836.
%H A332510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>
%F A332510 G.f.: (1/(1 - x)) * ((theta_3(x) - 1) / 2 - Sum_{k>=2} lambda(k-1) * x^k / (1 - x^k)).
%F A332510 a(n) = floor(sqrt(n)) - Sum_{k=1..n} Sum_{d|k, d > 1} lambda(d-1).
%F A332510 Sum_{k=1..n} mu(k) * a(floor(n/k)) = lambda(n).
%t A332510 Table[Sum[LiouvilleLambda[Floor[n/k]], {k, 1, n}], {n, 1, 85}]
%t A332510 Table[Floor[Sqrt[n]] - Sum[DivisorSum[k, LiouvilleLambda[# - 1] &, # > 1 &], {k, 1, n}], {n, 1, 85}]
%t A332510 nmax = 85; CoefficientList[Series[(1/(1 - x)) ((EllipticTheta[3, 0, x] - 1)/2 - Sum[LiouvilleLambda[k - 1] x^k/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x] // Rest
%o A332510 (PARI) a(n) = sum(k=1, n, (-1)^bigomega(n\k)); \\ _Michel Marcus_, Feb 14 2020
%Y A332510 Cf. A000196, A002819, A006218, A008683, A008836, A317625, A332509.
%K A332510 nonn
%O A332510 1,4
%A A332510 _Ilya Gutkovskiy_, Feb 14 2020