A333810 -id:A333810 - cp's OEIS frontend

A333807 Sum of odd divisors of n that are < sqrt(n).

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 8, 4, 1, 1, 9, 1, 1, 11, 1, 6, 4, 1, 1, 4, 13, 1, 4, 1, 1, 9, 1, 8, 4, 1, 6, 4, 1, 1, 11, 6, 1, 4, 1, 1, 18

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

Cf. A000593, A069289, A070039, A333805, A333808, A333810.

Table[DivisorSum[n, # &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[(2 k - 1) x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

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

sign

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

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

Original entry on oeis.org

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

sign

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

A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

Original entry on oeis.org

0, 1, -1, 1, -1, -1, -1, 3, -1, -1, -1, 6, -1, -1, -4, 3, -1, 2, -1, -1, -4, -1, -1, 10, -1, -1, -4, -1, -1, 7, -1, 7, -4, -1, -6, 2, -1, -1, -4, 12, -1, -4, -1, -1, -9, -1, -1, 16, -1, 4, -4, -1, -1, -4, -6, 14, -4, -1, -1, 13, -1, -1, -11, 7, -6, -4, -1, -1, -4, 11, -1, 8, -1, -1, -9, -1, -8, -4, -1, 20

Offset: 1

Ilya Gutkovskiy, Nov 04 2021

sign

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

Cf. A000593, A070039, A109506, A333810, A348660, A348951, A348952, A348953, A348955, A348956.

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

A348954(n) = sumdiv(n,d,if((d*d)Antti Karttunen, Nov 05 2021

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

A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, -3, 1, -3, 1, -3, 1, 6, 1, -3, 10, -3, 1, 6, 1, -19, 10, -3, 1, -10, 1, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -10, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 1, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -19, 26, -30, 1, -19, 10, 71

Offset: 1

Ilya Gutkovskiy, May 20 2024

sign

Cf. A056924, A064027, A070039, A321543, A333809, A333810, A339353, A344300, A372625, A373031.

nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k (k + 1))/(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

A333807 Sum of odd divisors of n that are < sqrt(n).

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

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

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A348954 a(n) = Sum_{d|n, d < sqrt(n)} (-1)^(n/d) * d.

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PARI

Formula

A373032 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k*(k+1)) / (1 - x^k).

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