A048272 -id:A048272 - cp's OEIS frontend

A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

0, 1, 2, 6, 11, 22, 40, 70, 116, 191, 304, 474, 726, 1094, 1624, 2384, 3453, 4950, 7030, 9890, 13798, 19108, 26264, 35858, 48652, 65615, 87996, 117396, 155826, 205854, 270728, 354506, 462306, 600544, 777184, 1002180, 1287889, 1649578, 2106152, 2680924

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A209423 and A000009.
Convolution of A015723 and A000041.
Convolution of A048272 and A015128.
a(n) is the number of overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

Cf. A006128, A015723, A209423, A305082, A305102.

Cf. A335651 and A335666.

nmax = 40; CoefficientList[Series[Sum[x^k/(1+x^k), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, 1*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (4*Pi*sqrt(n)).

a(n) = A305122(n) + A305124(n).

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

sign

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

A307397 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^kA(x)^k/(1 + x^kA(x)^k).

Original entry on oeis.org

1, 1, 1, 3, 8, 18, 50, 150, 429, 1258, 3835, 11740, 36148, 112856, 355318, 1124582, 3582186, 11477162, 36939043, 119387415, 387393424, 1261422550, 4120343870, 13498085604, 44337516318, 145993301239, 481812344551, 1593439356575, 5280074015618, 17528034861180

Offset: 0

Ilya Gutkovskiy, Apr 07 2019

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 18 x^5 + 50*x^6 + 150*x^7 + 429*x^8 + 1258*x^9 + 3835*x^10 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1835

Cf. A048272, A190790, A192206, A307399, A307401.

terms = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k A[x]^k/ (1 + x^k A[x]^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; A[] = 0; Do[A[x] = 1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Sum[(-1)^(d + 1), {d, Divisors[k]}] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
(* Calculation of constant d: *) val = r /. FindRoot[{Log[1 - r*s] + QPolyGamma[0, 1 - I*Pi/Log[r*s], r*s] == (1 - s)*Log[r*s], (1 - s)/s - I*Pi*QPolyGamma[1, 1 - I*Pi/Log[r*s], r*s] / (s*Log[r*s]^2) + r*(1/(1 - r*s) - Derivative[0, 0, 1][QPolyGamma][0, 1 - I*Pi/Log[r*s], r*s]) == Log[r*s]}, {r, 1/3}, {s, 2}, WorkingPrecision -> 100]; N[1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Sep 27 2023 *)

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} A048272(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} A048272(k)*x^k)).
a(n) ~ c * d^n / n^(3/2), where d = 3.494393309755265712092948136162079492013891957890570364999827377394989... and c = 0.4966863488644340281558816065879601408815044380350316600850227488... - Vaclav Kotesovec, Sep 27 2023

A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

Antti Karttunen, Table of n, a(n) for n = 1..11025
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000290, A010052, A046951, A048272, A113024, A300853.

seq(coeff(series(add(x^(k^2)/(1+x^(k^2)),k=1..n), x,n+1),x,n),n=1..100); # Muniru A Asiru, Jul 30 2018

nmax = 95; Rest[CoefficientList[Series[Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + x^k^2)^(1/k^2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, IntegerQ[#^(1/2)] &], {n, 95}]
f[p_, e_] := If[p == 2, If[OddQ[e], -Floor[e/2 + 1], -Floor[(e - 1)/2]], Floor[e/2 + 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

A317529(n) = sumdiv(n,d,((-1)^(1+(n/d)))*issquare(d)); \\ Antti Karttunen, Nov 07 2018

G.f.: Sum_{k>=1} x^A000290(k)/(1 + x^A000290(k)).
L.g.f.: log(Product_{k>=1} (1 + x^(k^2))^(1/k^2)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010052(d).
If n is odd, a(n) = A046951(n).
Multiplicative with a(2^e) = -floor(e/2+1) for odd e, -floor((e-1)/2) for even e, and a(p^e) = floor(e/2+1) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s) * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = -(sqrt(2)-1) * zeta(1/2) = 0.604898... (A113024). (End)

A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, -1, 2, 1, 1, -1, 1, 1, 3, -2, 1, 1, 1, -1, 3, 1, 1, -3, 2, 1, 3, -1, 1, 1, 1, -3, 3, 1, 3, -2, 1, 1, 3, -3, 1, 1, 1, -1, 5, 1, 1, -5, 2, 1, 3, -1, 1, 1, 3, -3, 3, 1, 1, -3, 1, 1, 5, -4, 3, 1, 1, -1, 3, 1, 1, -5, 1, 1, 5, -1, 3, 1, 1, -5

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Number of odd proper divisors of n minus number of even proper divisors of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A032741, A048272, A058344, A091954, A275495 (partial sums), A325939.

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

A325937(n) = -sumdiv(n, d, if(d==n,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

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

a(n) = -Sum_{d|n, d

a(n) = A048272(n) + (-1)^n.

A332682 a(n) = Sum_{k=1..n} (-1)^(k+1) * ceiling(n/k).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 6, 5, 7, 8, 9, 8, 9, 10, 13, 11, 12, 13, 14, 13, 16, 17, 18, 15, 17, 18, 21, 20, 21, 22, 23, 20, 23, 24, 27, 25, 26, 27, 30, 27, 28, 29, 30, 29, 34, 35, 36, 31, 33, 34, 37, 36, 37, 38, 41, 38, 41, 42, 43, 40, 41, 42, 47, 43, 46, 47, 48, 47, 50

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006590, A048272, A059851, A275495, A325937.

f:= proc(n) local k; add((-1)^(k+1)*ceil(n/k),k=1..n) end proc:
map(f, [$1..100]); # Robert Israel, Nov 25 2024

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

a(n) = sum(k=1, n, (-1)^(k+1)*ceil(n/k)); \\ Michel Marcus, Feb 21 2020

G.f.: (x/(1 - x)) * (1 + Sum_{k>=2} x^k / (1 + x^k)).
G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} (-1)^(k+1) * x^(2*k) / (1 - x^k)).
a(n) = (n mod 2) + Sum_{k=1..n-1} A048272(k).
a(n) = 1 + Sum_{k<=n-1} A325937(k). - Robert Israel, Nov 25 2024

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

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, A333809, A258998, A348951.

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

From Ridouane Oudra, Sep 04 2025: (Start)
a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d+1).
a(n) = A333809(n) + A258998(n).
a(n) = A348951(n) + A048272(n). (End)

A338814 Expansion of e.g.f. log(Product_{k>0} (1 + x^k)^(1/k)).

Original entry on oeis.org

1, 0, 4, -6, 48, 0, 1440, -10080, 120960, 0, 7257600, -79833600, 958003200, 0, 348713164800, -3923023104000, 41845579776000, 0, 12804747411456000, -243290200817664000, 9731608032706560000, 0, 2248001455555215360000, -103408066955539906560000

Offset: 1

Seiichi Manyama, Nov 10 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 1..450

Column 1 of A338813.

Cf. A048272, A168243, A318249.

a[n_] := (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]; Array[a, 25] (* Amiram Eldar, Apr 28 2021 *)

N=40; x='x+O('x^N); Vec(serlaplace(log(prod(k=1, N, (1+x^k)^(1/k)))))

{a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)))}

a(n) = (n-1)! * A048272(n).

A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 14 2021

Number of odd squares dividing n minus number of even squares dividing n.

Inverse Moebius transform of A258998.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046951, A048272, A072691, A258998, A317529, A344300.

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
f[p_, e_] := 1 - (-1)^p*Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 - (-1)^f[i,1] * floor(f[i,2]/2));} \\ Amiram Eldar, Nov 15 2022

G.f.: Sum_{k>=1} (1 - theta_4(x^k)) / 2.
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(2^e) = 1 - floor(e/2), and a(p^e) = 1 + floor(e/2) for p > 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 (A072691). (End)

A357761 a(n) = A227872(n) - A356018(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 2, 4, -1, 0, 2, 0, 2, 4, -2, 5, 0, -2, 2, 0, 2, 4, 0, 0, 1, 4, -2, 6, 0, -4, 2, 6, 0, 0, 2, -3, 2, 4, 0, 0, 2, 4, 0, 6, -4, 0, 2, 0, 3, 2, -2, 6, 0, -4, 2, 8, 0, 0, 2, -6, 2, 4, 0, 7, 0, 0, 2, 0, 0, 4, 0, -4, 2, 4, -2, 6, 2, 0, 2, 0, -1, 4, 0

Offset: 1

Amiram Eldar, Oct 12 2022

The excess of the number of odious (A000069) divisors of n over the number of evil (A001969) divisors of n.

Every integer occurs in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000069, A000290 (positions of odd terms), A001969, A027697, A027699, A106400, A227872, A230851 (positions of 0's), A356018, A357762.

Similar sequences: A046660, A048272.

a[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; Array[a, 100]

a(n) = -sumdiv(n, d, (-1)^hammingweight(d));

a(n) = -Sum_{d|n} A106400(d).
a(n) = A000005(n) - 2*A356018(n).
a(n) = 2*A227872(n) - A000005(n).
a(n) = 0 iff n is in A230851.
a(n) == 1 (mod 2) iff n is a square (A000290).
a(2^n) = n + 1.
a(p*2^n) = 0 when p is an evil prime (A027699).
a(p^2*2^n) = n + 1 when p is an evil prime (A027699) and p^2 is odious, and when p is an odd odious prime (A027697) and p^2 is evil.
a(p^2*2^n) = -(n+1) when p is an evil prime and p^2 is also evil.
a(p^2*2^n) = 3*(n+1) when p is an odd odious prime and p^2 is also odious.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Sum_{k>=1} A106400(k)/k = 1.196283264... (A357762).

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A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

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

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A307397 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k*A(x)^k).

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A317529 Expansion of Sum_{k>=1} x^(k^2)/(1 + x^(k^2)).

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A325937 Expansion of Sum_{k>=1} (-1)^(k + 1) * x^(2*k) / (1 - x^k).

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A332682 a(n) = Sum_{k=1..n} (-1)^(k+1) * ceiling(n/k).

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

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A338814 Expansion of e.g.f. log(Product_{k>0} (1 + x^k)^(1/k)).

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A307397 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^kA(x)^k/(1 + x^kA(x)^k).