A348608 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

1, -1, 1, 1, 1, -3, 1, 1, 4, -3, 1, -2, 1, -3, 4, 5, 1, -6, 1, -3, 4, -3, 1, 2, 6, -3, 4, -3, 1, -11, 1, 5, 4, -3, 6, 0, 1, -3, 4, 0, 1, -12, 1, -3, 9, -3, 1, 8, 8, -8, 4, -3, 1, -12, 6, -2, 4, -3, 1, -5, 1, -3, 11, 13, 6, -12, 1, -3, 4, -15, 1, 0, 1, -3, 9, -3, 8, -12, 1, 8

Offset: 1

Author

Ilya Gutkovskiy, Oct 25 2021

Keywords

• sign
• changed

Links

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

Crossrefs

Cf. A000593, A006005, A046897, A066839, A109506, A305152, A333782, A037213, A348953.

Programs

Mathematica

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

PARI

a(n) = sumdiv(n, d, if (d<=sqrt(n), (-1)^(d + n/d)*d)); \\ Michel Marcus, Oct 25 2021

Formula

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

a(n) = 1 if n = 1 or n is an odd prime (A006005) or n = 4 or n = 8. - Bernard Schott, Dec 18 2021

a(n) = A037213(n) - A348953(n). - Ridouane Oudra, Aug 21 2025