A348660 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1) * d.

1, -1, 1, -3, 1, 1, 1, -3, 4, 1, 1, -6, 1, 1, 4, -7, 1, -2, 1, 1, 4, 1, 1, -10, 6, 1, 4, 1, 1, -7, 1, -7, 4, 1, 6, -8, 1, 1, 4, -12, 1, 4, 1, 1, 9, 1, 1, -16, 8, -4, 4, 1, 1, 4, 6, -14, 4, 1, 1, -13, 1, 1, 11, -15, 6, 4, 1, 1, 4, -11, 1, -8, 1, 1, 9, 1, 8, 4, 1, -20

Offset: 1

Author

Ilya Gutkovskiy, Oct 28 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000593, A066839, A109506, A333782, A348608.

Programs

Mathematica

Table[DivisorSum[n, (-1)^(n/# + 1) # &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, if(sqr(d) <= n, (-1)^(n/d + 1)*d, 0)); \\ Michel Marcus, Oct 28 2021, corrected by Antti Karttunen, Dec 14 2021

Formula

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