A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.

1, 3, 9, 12, 22, 30, 44, 48, 71, 83, 105, 115, 141, 157, 201, 206, 240, 266, 304, 318, 378, 402, 448, 460, 519, 547, 623, 641, 699, 747, 809, 815, 907, 943, 1035, 1064, 1138, 1178, 1286, 1302, 1384, 1448, 1534, 1560, 1710, 1758, 1852, 1866, 1977, 2039, 2179, 2209, 2315, 2395, 2535

Offset: 1

Author

Seiichi Manyama, May 27 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=2 of A344726.

Cf. A059851, A078471, A222548.

Programs

Mathematica

a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-2*DivisorSigma[0, 2*n] + 3*DivisorSigma[0, n] + 2*DivisorSigma[1, 2*n] - 4*DivisorSigma[1, n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)

PARI

a(n) = sum(k=1, n, (-1)^(k+1)*(n\k)^2);

PARI

a(n) = sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*(2*d-1)));

PARI

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^k))/(1-x))

Formula

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (2*d - 1).

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

a(n) ~ Pi^2 * n^2 / 12. - Vaclav Kotesovec, May 28 2021