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A211270 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y = 2n.

%I A211270 #22 Feb 06 2024 01:41:37
%S A211270 0,1,1,1,1,2,1,2,2,2,1,3,1,2,3,2,1,4,1,3,3,2,1,4,2,2,3,3,1,5,1,3,3,2,
%T A211270 3,5,1,2,3,4,1,5,1,3,5,2,1,5,2,4,3,3,1,5,3,4,3,2,1,7,1,2,5,3,3,5,1,3,
%U A211270 3,5,1,7,1,2,5,3,3,5,1,5,4,2,1,7,3,2,3,4,1,8,3,3,3,2,3,6,1,4,5
%N A211270 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y = 2n.
%C A211270 For a guide to related sequences, see A211266.
%H A211270 Antti Karttunen, <a href="/A211270/b211270.txt">Table of n, a(n) for n = 1..65537</a>
%H A211270 David J. Hemmer and Karlee J. Westrem, <a href="https://arxiv.org/abs/2402.02250">Palindrome Partitions and the Calkin-Wilf Tree</a>, arXiv:2402.02250 [math.CO], 2024. See Theorem 4.2 p. 7.
%F A211270 a(n) = floor((A000005(2n)-1)/2). - _Robert Israel_, Feb 25 2019
%e A211270 a(12) counts these pairs: (2,12), (3,8), (4,6).
%e A211270 For n = 2, only the pair (2,2) satisfies the condition, thus a(2) = 1. - _Antti Karttunen_, Sep 30 2018
%p A211270 seq(floor((numtheory:-tau(2*n)-1)/2),n=1..100); # _Robert Israel_, Feb 25 2019
%t A211270 a = 1; b = n; z1 = 120;
%t A211270 t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
%t A211270 {y, x, b}]]
%t A211270 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A211270 Table[c[n, n], {n, 1, z1}]           (* A038548 *)
%t A211270 Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
%t A211270 Table[c[n, 2*n], {n, 1, z1}]         (* this sequence *)
%t A211270 Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
%t A211270 Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
%t A211270 c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
%t A211270 Print
%t A211270 Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
%t A211270 Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
%t A211270 Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
%t A211270 Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
%t A211270 Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)
%o A211270 (PARI) A211270(n) = sumdiv(2*n,y,(((2*n/y)<=y)&&(y<=n))); \\ _Antti Karttunen_, Sep 30 2018
%Y A211270 Cf. A000005, A211266, A211261.
%K A211270 nonn
%O A211270 1,6
%A A211270 _Clark Kimberling_, Apr 07 2012
%E A211270 Term a(2) corrected by _Antti Karttunen_, Sep 30 2018