cp's OEIS Frontend

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

%I A363341 #35 Aug 04 2025 02:46:19
%S A363341 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,
%T A363341 17,16,17,17,20,21,20,20,23,22,21,22,24,23,26,25,28,27,26,25,27,29,30,
%U A363341 31,32,31,32,31,32,33,34,33,35,34,37,37,40,39,38,39,40
%N A363341 Number of positive integers k <= n such that round(n/k) is odd.
%C A363341 Here round(x) = floor(x + 1/2).
%C A363341 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)
%H A363341 Robert Israel, <a href="/A363341/b363341.txt">Table of n, a(n) for n = 1..10000</a>
%H A363341 Nicholas Dent and Caleb M. Shor, <a href="https://arxiv.org/abs/2206.15452">On residues of rounded shifted fractions with a common numerator</a>, arXiv:2206.15452 [math.NT], 2022.
%F A363341 a(n) = n - floor(2n/3) + floor(2n/5) - floor(2n/7) + ...
%F A363341 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.
%e A363341 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.
%p A363341 f:= proc(n) local k;
%p A363341    nops(select(k -> floor(n/k + 1/2)::odd, [$1..n]))
%p A363341 end proc:
%p A363341 map(f, [$1..120]); # _Robert Israel_, Aug 03 2025
%o A363341 (PARI) a(n) = sum(k=1, n, round(n/k)%2) \\ _Andrew Howroyd_, May 28 2023
%Y A363341 Cf. A059851 (number of k=1..n such that floor(n/k) is odd).
%Y A363341 Cf. A330926 (number of k=1..n such that ceiling(n/k) is odd).
%Y A363341 Cf. A057655 (number of lattice points in circle).
%Y A363341 Cf. A001826 (d_1), A001842 (d_3), A002654 (d_1-d_3).
%Y A363341 Cf. A077024 (n + floor(2n/3) + floor(2n/5) + floor(2n/7) + ...).
%K A363341 nonn
%O A363341 1,3
%A A363341 _Caleb M. Shor_, May 28 2023