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

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, 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, 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

Offset

1,6

Links

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

Eric Weisstein's World of Mathematics, Moebius Function

Formula

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

a(n) = 1 - Sum_{k=1..n} Sum_{d|k, d > 1} mu(d-1) for n > 0.

Sum_{k=1..n-1} mu(k) * a(floor(n/k)) = 0.

Mathematica

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

Prog

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

Crossrefs

Cf. A002321, A006218, A008683, A098018, A317625, A332510.

Sequence in context: A364663 A101050 A128979 * A336157 A190167 A339010

Adjacent sequences: A332506 A332507 A332508 * A332510 A332511 A332512

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 14 2020

Status

approved