A332681 - OEIS

%I #7 Feb 21 2020 11:03:30

%S 1,3,4,8,11,20,23,35,43,56,63,83,90,115,128,144,159,191,202,238,255,

%T 280,299,343,359,400,424,460,483,538,553,613,646,687,720,768,791,864,

%U 901,949,980,1059,1082,1166,1206,1255,1298,1390,1422,1506,1547,1611,1658,1762

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

%F 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).

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

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

%t Table[Sum[MoebiusMu[k] Ceiling[n/k]^2, {k, 1, n}], {n, 1, 54}]

%t Join[{1}, Table[2 + Sum[2 EulerPhi[k - 1] + MoebiusMu[k], {k, 2, n}], {n, 2, 54}]]

%o (PARI) a(n) = sum(k=1, n, moebius(k)*ceil(n/k)^2); \\ _Michel Marcus_, Feb 21 2020

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

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Feb 19 2020