A330926 -id:A330926 - cp's OEIS frontend

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

1, 2, 4, 4, 8, 8, 11, 11, 19, 16, 21, 21, 30, 30, 37, 29, 45, 45, 51, 51, 66, 56, 67, 67, 88, 83, 96, 84, 105, 105, 112, 112, 144, 130, 147, 135, 159, 159, 178, 162, 197, 197, 208, 208, 241, 209, 232, 232, 277, 270, 290, 270, 309, 309, 324, 308, 357, 335, 364, 364

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

Cf. A000217, A000593, A078471, A120885, A330926, A332490.

b:= n-> add(d, d=select(x-> x::odd, numtheory[divisors](n))):
a:= proc(n) option remember; n+`if`(n<2, 0, a(n-1))-b(n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2] k, {k, 1, n}], {n, 1, 60}]
Table[n (n + 1)/2 - Sum[DivisorSum[k, (-1)^(k/# + 1) # &], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x) (1/(1 - x)^2 - Sum[k x^k/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (ceil(n/k) % 2)*k); \\ Michel Marcus, May 26 2020

G.f.: (x/(1 - x)) * (1/(1 - x)^2 - Sum_{k>=1} k * x^k / (1 + x^k)).
a(n) = n*(n + 1)/2 - Sum_{k=1..n-1} A000593(k).
a(n) = A000217(n) - A078471(n-1).

A363341 Number of positive integers k <= n such that round(n/k) is odd.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 4, 6, 7, 6, 5, 9, 8, 9, 9, 10, 10, 11, 12, 13, 12, 13, 12, 15, 16, 17, 16, 17, 16, 17, 17, 20, 21, 20, 20, 23, 22, 21, 22, 24, 23, 26, 25, 28, 27, 26, 25, 27, 29, 30, 31, 32, 31, 32, 31, 32, 33, 34, 33, 35, 34, 37, 37, 40, 39, 38, 39, 40

Offset: 1

Caleb M. Shor, May 28 2023

nonn

Here round(x) = floor(x + 1/2).

a(n) is related to the number of lattice points in a circle. Let C(x) equal the number of square lattice points in a circle of radius sqrt(x) centered at the origin. Then a(n) = (C(2n) - 4n - 1)/4. (Prop 3.5 in Dent & Shor paper)

			For n=5: round(5/1), round(5/2), round(5/3), round(5/4), round(5/5) = 5, 3, 2, 1, 1 among which 4 are odd so a(5)=4.

Robert Israel, Table of n, a(n) for n = 1..10000
Nicholas Dent and Caleb M. Shor, On residues of rounded shifted fractions with a common numerator, arXiv:2206.15452 [math.NT], 2022.

Cf. A059851 (number of k=1..n such that floor(n/k) is odd).
Cf. A330926 (number of k=1..n such that ceiling(n/k) is odd).
Cf. A057655 (number of lattice points in circle).
Cf. A001826 (d_1), A001842 (d_3), A002654 (d_1-d_3).
Cf. A077024 (n + floor(2n/3) + floor(2n/5) + floor(2n/7) + ...).

f:= proc(n) local k;
   nops(select(k -> floor(n/k + 1/2)::odd, [$1..n]))
end proc:
map(f, [$1..120]); # Robert Israel, Aug 03 2025

a(n) = sum(k=1, n, round(n/k)%2) \\ Andrew Howroyd, May 28 2023

a(n) = n - floor(2n/3) + floor(2n/5) - floor(2n/7) + ...

a(n) = -n + Sum_{k=1..2n} d_1(k) - d_3(k), where d_i(k) is the number of divisors of k that are congruent to i modulo 4.

cp's OEIS Frontend

A333194 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2) * k.

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