A332510 - OEIS

A332510

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

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, 8, 5, 10, 9, 10, 11, 10, 9, 12, 13, 12, 13, 12, 11, 16, 17, 12, 13, 12, 13, 16, 13, 14, 15, 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

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OFFSET

1,4

LINKS

Table of n, a(n) for n=1..85.

Eric Weisstein's World of Mathematics, Liouville Function

FORMULA

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

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

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

MATHEMATICA

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

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

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

PROG

(PARI) a(n) = sum(k=1, n, (-1)^bigomega(n\k)); \\ Michel Marcus, Feb 14 2020

CROSSREFS

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

Sequence in context: A099986 A166478 A050325 * A001314 A020733 A210700

Adjacent sequences: A332507 A332508 A332509 * A332511 A332512 A332513

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Feb 14 2020

STATUS

approved