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

%I #8 Feb 15 2020 10:48:21

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>

%F G.f.: (1/(1 - x)) * ((theta_3(x) - 1) / 2 - Sum_{k>=2} lambda(k-1) * x^k / (1 - x^k)).

%F a(n) = floor(sqrt(n)) - Sum_{k=1..n} Sum_{d|k, d > 1} lambda(d-1).

%F Sum_{k=1..n} mu(k) * a(floor(n/k)) = lambda(n).

%t Table[Sum[LiouvilleLambda[Floor[n/k]], {k, 1, n}], {n, 1, 85}]

%t Table[Floor[Sqrt[n]] - Sum[DivisorSum[k, LiouvilleLambda[# - 1] &, # > 1 &], {k, 1, n}], {n, 1, 85}]

%t 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 (PARI) a(n) = sum(k=1, n, (-1)^bigomega(n\k)); \\ _Michel Marcus_, Feb 14 2020

%Y Cf. A000196, A002819, A006218, A008683, A008836, A317625, A332509.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Feb 14 2020

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Last modified March 28 13:35 EDT 2024. Contains 371254 sequences. (Running on oeis4.)