A211272 - OEIS

%I #14 Sep 08 2022 08:46:02

%S 0,1,1,1,1,1,1,2,2,1,1,2,2,1,1,2,2,2,2,2,2,1,1,3,3,1,1,2,2,2,2,3,3,1,

%T 1,3,3,1,1,3,3,2,2,2,2,1,1,4,4,2,2,2,2,2,2,3,3,1,1,4,4,1,1,3,3,2,2,2,

%U 2,2,2,5,5,1,1,2,2,2,2,4,4,1,1,4,4,1,1,3,3,3,3,2,2,1,1,5,5,2,2

%N Number of integer pairs (x,y) such that 0<x<=y<=n and x*y=floor(n/2).

%C For a guide to related sequences, see A211266.

%H Robert Israel, <a href="/A211272/b211272.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = ceiling(A000005(floor(n/2))/2). - _Robert Israel_, Feb 07 2020

%e a(24) counts these pairs: (1,12), (2,6), (3,4).

%p [seq(ceil(numtheory:-tau(floor(n/2))/2),n=1..100)]; - _Robert Israel_, Feb 07 2020

%t a = 1; b = n; z1 = 120;

%t t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},

%t {y, x, b}]]

%t c[n_, k_] := c[n, k] = Count[t[n], k]

%t Table[c[n, n], {n, 1, z1}]           (* A038548 *)

%t Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)

%t Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)

%t Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)

%t Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)

%t c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]

%t Print

%t Table[c1[n, n], {n, 1, z1}]          (* A094820 *)

%t Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)

%t Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)

%t Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)

%t Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

%o (Magma) [0] cat [Ceiling(#Divisors( Floor(n/2))/2):n in [2..100]]; // _Marius A. Burtea_, Feb 07 2020

%Y Cf. A000005, A211266.

%K nonn

%O 1,8

%A _Clark Kimberling_, Apr 07 2012