A332681 a(n) = Sum_{k=1..n} mu(k) * ceiling(n/k)^2.

1, 3, 4, 8, 11, 20, 23, 35, 43, 56, 63, 83, 90, 115, 128, 144, 159, 191, 202, 238, 255, 280, 299, 343, 359, 400, 424, 460, 483, 538, 553, 613, 646, 687, 720, 768, 791, 864, 901, 949, 980, 1059, 1082, 1166, 1206, 1255, 1298, 1390, 1422, 1506, 1547, 1611, 1658, 1762

Offset: 1

Author

Ilya Gutkovskiy, Feb 19 2020

Keywords

• nonn

Crossrefs

Cf. A000010, A002088, A002321, A008683, A006590, A018805, A206350, A332623.

Programs

Mathematica

Table[Sum[MoebiusMu[k] Ceiling[n/k]^2, {k, 1, n}], {n, 1, 54}]
Join[{1}, Table[2 + Sum[2 EulerPhi[k - 1] + MoebiusMu[k], {k, 2, n}], {n, 2, 54}]]

PARI

a(n) = sum(k=1, n, moebius(k)*ceil(n/k)^2); \\ Michel Marcus, Feb 21 2020

Formula

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

a(n) = 2 + Sum_{k=2..n} (2 * phi(k-1) + mu(k)) for n > 1.

a(n) = 1 + 2 * A002088(n-1) + A002321(n) for n > 1.