A325937 -id:A325937 - cp's OEIS frontend

A325939 Expansion of Sum_{k>=1} x^(2*k) / (1 + x^k).

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

Offset: 1

Ilya Gutkovskiy, Sep 09 2019

Number of even divisors of n minus number of odd strong divisors of n (i.e. odd divisors > 1).

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

Cf. A032741, A048272, A075997 (partial sums), A325937.

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

A325939(n) = sumdiv(n, d, if(1==d,0,((-1)^d))); \\ Antti Karttunen, Sep 20 2019

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

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

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A325939 Expansion of Sum_{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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A335021 a(n) = Sum_{d|n, 1 < d < n} (-1)^(d + 1).

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