A325939 -id:A325939 - cp's OEIS frontend

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

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.

A264400 Number of parts of even multiplicities in all the partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 6, 6, 15, 15, 29, 34, 58, 70, 109, 132, 199, 246, 348, 435, 601, 746, 1005, 1252, 1653, 2053, 2666, 3298, 4231, 5219, 6608, 8124, 10198, 12476, 15525, 18927, 23374, 28387, 34823, 42122, 51376, 61922, 75098, 90200, 108874, 130298, 156564, 186777, 223490, 265779, 316799

Offset: 0

Emeric Deutsch, Nov 21 2015

nonn

a(n) = Sum_{k>=0} k*A264399(n,k).

			a(6) = 6 because we have [6], [5,1], [4,2], [4,1*,1], [3*,3], [3,2,1], [3,1,1,1], [2,2,2], [2*,2,1*,1], [2,1*,1,1,1], and [1*,1,1,1,1,1] (the 6 parts with even multiplicities are marked).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A264399, A325939.

g := (sum(x^(2*j)/(1+x^j), j = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 0 .. 60);

Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* Michael De Vlieger, Nov 21 2015 *)
Table[Sum[(1 - 2*DivisorSigma[0, 2*k] + 3*DivisorSigma[0, k]) * PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* Vaclav Kotesovec, Jun 14 2025 *)

{ my(n=50); Vec(sum(k=1, n, x^(2*k)/(1+x^k) + O(x*x^n)) / prod(k=1, n, 1-x^k + O(x*x^n)), -(n+1)) } \\ Andrew Howroyd, Dec 22 2017

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

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020

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

a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020

from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).
a(n) = n - Sum_{k=1..n-1} A048272(k).
a(n) = A075997(n-1) + 1.

A335021 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 19 2020

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

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A048272, A070824, A325937, A325939, A335022.

Cf. A001227, A007814.

Table[DivisorSum[n, (-1)^(# + 1) &, 1 < # < n &], {n, 1, 88}]
nmax = 88; CoefficientList[Series[Sum[(-1)^(k + 1) x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d>1) && (dMichel Marcus, May 20 2020

from sympy import divisor_count
def A335021(n): return 0 if n == 1 else (1-(m:=(~n & n-1).bit_length()))*divisor_count(n>>m)-((n&1)<<1) # Chai Wah Wu, Jul 01 2022

G.f.: Sum_{k>=2} (-1)^(k + 1) * x^(2*k) / (1 - x^k).
G.f.: - Sum_{k >= 2} x^(2*k)/(1 + x^k). - Peter Bala, Jan 12 2021
a(n) = A001227(n)*(1 - A007814(n)) - 1 + (-1)^n, if n > 1. - Sebastian Karlsson, Jan 14 2021

Showing 1-4 of 4 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A264400 Number of parts of even multiplicities in all the partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A335021 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula