A348515 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(n/d + 1).

1, -1, 1, -2, 1, 0, 1, -2, 2, 0, 1, -3, 1, 0, 2, -3, 1, -1, 1, -1, 2, 0, 1, -4, 2, 0, 2, -1, 1, -2, 1, -3, 2, 0, 2, -3, 1, 0, 2, -4, 1, 0, 1, -1, 3, 0, 1, -5, 2, -1, 2, -1, 1, 0, 2, -4, 2, 0, 1, -4, 1, 0, 3, -4, 2, 0, 1, -1, 2, -2, 1, -4, 1, 0, 3, -1, 2, 0, 1, -5

Offset: 1

Author

Ilya Gutkovskiy, Nov 04 2021

Keywords

• sign

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A038548, A048272, A113652, A228441, A305152, A333781, A348660, A348951, A348952, A258998.

Programs

Mathematica

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

PARI

A348515(n) = sumdiv(n,d,if((d*d)<=n,(-1)^(1 + (n/d)),0)); \\ Antti Karttunen, Nov 05 2021

Python

from sympy import divisors
def a(n): return sum((-1)**(n//d + 1) for d in divisors(n) if d*d <= n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 22 2021

Formula

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

a(n) = 1 iff n = 1 or n is an odd prime (A006005). - Bernard Schott, Nov 22 2021

a(n) = A258998(n) - A348951(n). - Ridouane Oudra, Aug 21 2025