A348608 -id:A348608 - cp's OEIS frontend

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

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

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

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

Cf. A000593, A046897, A070039, A109506, A333810, A348608, A348951, A348952, A348954, A348955, A348956, A037213.

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

A348953(n) = -sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

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

a(n) = A037213(n) - A348608(n). - Ridouane Oudra, Aug 21 2025

A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, May 26 2018

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Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A038548, A048272, A193773 (odd bisection), A348608, A228441, A010052, A348952.

{a(n) = polcoeff(sum(k=1, sqrtint(n), x^(k^2)/(1+x^k))+x*O(x^n), n)}

a(n) = sumdiv(n, d, if (d <= sqrtint(n), (-1)^(d + n/d))); \\ Michel Marcus, Nov 03 2021

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d). - Ilya Gutkovskiy, Nov 02 2021
a(n) = (A228441(n) + A010052(n))/2. - Ridouane Oudra, Aug 14 2025
a(n) = A010052(n) - A348952(n). - Ridouane Oudra, Aug 20 2025

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).

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79

Offset: 1

Ilya Gutkovskiy, May 07 2024

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Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

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

cp's OEIS Frontend

A348953 a(n) = -Sum_{d|n, d < sqrt(n)} (-1)^(d + n/d) * d.

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A305152 Expansion of Sum_{k>0} x^(k^2) / (1 + x^k).

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

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Author

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Mathematica

PARI

Formula

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula