A333782 -id:A333782 - cp's OEIS frontend

A333810 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k*(k + 1)) / (1 - x^k).

0, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 2, 1, -1, 4, -1, 1, 2, 1, -5, 4, -1, 1, -2, 1, -1, 4, -5, 1, 7, 1, -5, 4, -1, 6, -2, 1, -1, 4, 0, 1, -4, 1, -5, 9, -1, 1, -8, 1, 4, 4, -5, 1, -4, 6, 2, 4, -1, 1, -3, 1, -1, 11, -5, 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, A070039, A333782, A333807, A333809.

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

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

A333781 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 - x^k).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, -1, 1, 1, 1, -1, 2, 0, 1, 0, 2, 0, 2, -1, 1, 2, 1, -1, 2, 0, 2, -1, 1, 0, 2, 0, 1, 0, 1, -1, 3, 0, 1, -1, 2, 1, 2, -1, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 3, -2, 2, 0, 1, -1, 2, 2, 1, -2, 1, 0, 3, -1, 2, 0, 1, -1, 3, 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. A038548, A048272, A069288, A333782.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) 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, 1, -1))); \\ Michel Marcus, Apr 05 2020

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Oct 28 2021

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

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

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

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

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

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.

cp's OEIS Frontend

A333810 G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^(k*(k + 1)) / (1 - x^k).

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

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A333781 G.f.: Sum_{k>=1} (-1)^(k + 1) * x^(k^2) / (1 - x^k).

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Author

Keywords

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Programs

Mathematica

PARI

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

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Mathematica

PARI

Formula

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

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Mathematica

Formula