A333781 -id:A333781 - cp's OEIS frontend

A333782 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k^2) / (1 - x^k).

1, 1, 1, -1, 1, -1, 1, -1, 4, -1, 1, 2, 1, -1, 4, -5, 1, 2, 1, -5, 4, -1, 1, -2, 6, -1, 4, -5, 1, 7, 1, -5, 4, -1, 6, -8, 1, -1, 4, 0, 1, -4, 1, -5, 9, -1, 1, -8, 8, 4, 4, -5, 1, -4, 6, 2, 4, -1, 1, -3, 1, -1, 11, -13, 6, -4, 1, -5, 4, 11, 1, -16, 1, -1, 9, -5, 8, -4, 1, -8

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

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Excess of sum of odd divisors of n that are <= sqrt(n) over sum of even divisors of n that are <= sqrt(n).

Cf. A002129, A066839, A069289, A333781.

nmax = 80; CoefficientList[Series[Sum[(-1)^(k + 1) k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if (d^2<=n, if (d%2, d, -d))); \\ Michel Marcus, Apr 05 2020

A333809 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

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Number of odd divisors of n that are < sqrt(n) minus number of even divisors of n that are < sqrt(n).

Cf. A048272, A056924, A333781, A333805, A333810.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

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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

A373031 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^k).

Original entry on oeis.org

1, 1, 1, -3, 1, -3, 1, -3, 10, -3, 1, 6, 1, -3, 10, -19, 1, 6, 1, -19, 10, -3, 1, -10, 26, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -46, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 50, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -83, 26, -30, 1, -19, 10, 71

Offset: 1

Ilya Gutkovskiy, May 20 2024

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Cf. A038548, A064027, A066839, A095118, A321543, A333781, A333782, A344300, A372625, A373032.

nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

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A333782 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k^2) / (1 - x^k).

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A333809 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k*(k + 1)) / (1 - x^k).

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

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A373031 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^k).

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