A305152 -id:A305152 - cp's OEIS frontend

A348952 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d).

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

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

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

Cf. A048272, A056924, A228441, A258453, A305152, A333809, A348515, A348951, A348953, A348954, A348955, A348956, A010052.

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

A348952(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

G.f.: Sum_{k>=1} x^(k*(k + 1)) / (1 + x^k).
For p odd prime, a(p) = a(p^2) = -1. - Bernard Schott, Nov 22 2021
a(n) = (A010052(n) - A228441(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A305152(n). - Ridouane Oudra, Aug 20 2025

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Oct 25 2021

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

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

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

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

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

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Nov 04 2021

sign

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

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

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

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

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

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

cp's OEIS Frontend

A348952 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d).

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

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

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