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A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.

%I A211264 #28 Sep 30 2024 12:42:15
%S A211264 0,1,2,3,4,6,7,9,10,12,13,16,17,19,21,23,24,27,28,31,33,35,36,40,41,
%T A211264 43,45,48,49,53,54,57,59,61,63,67,68,70,72,76,77,81,82,85,88,90,91,96,
%U A211264 97,100,102,105,106,110,112,116,118,120,121,127,128,130,133,136
%N A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.
%C A211264 Partial sums of A056924.
%C A211264 For a guide to related sequences, see A211266.
%H A211264 Robert Israel, <a href="/A211264/b211264.txt">Table of n, a(n) for n = 1..10000</a>
%F A211264 a(n) = (1/2)*Sum_{i=1..n} (1 - A008836(i))*floor(n/i). - _Enrique Pérez Herrero_, Jul 10 2012 [Corrected by _Ridouane Oudra_, Oct 17 2019]
%F A211264 From _Ridouane Oudra_, Oct 17 2019: (Start)
%F A211264 a(n) = Sum_{i=1..n} A066829(i)*floor(n/i)
%F A211264 a(n) = (1/2)*(A006218(n) - A000196(n)). (End)
%F A211264 From _Ridouane Oudra_, Sep 28 2024: (Start)
%F A211264 a(n) = Sum_{k=1..n} floor((sqrt(k^2 + 4*n) - k)/2) ;
%F A211264 a(n) = A094820(n) - A000196(n) ;
%F A211264 a(n) = A181972(2*n). (End)
%p A211264 with(numtheory): seq(add((bigomega(i) mod 2)*floor(n/i), i=1..n), n=1..60); # _Ridouane Oudra_, Oct 17 2019
%p A211264 # Alternative:
%p A211264 ListTools:-PartialSums(map(t-> floor(numtheory:-tau(t)/2), [$1..100])); # _Robert Israel_, Oct 18 2019
%t A211264 a = 1; b = n; z1 = 120;
%t A211264 t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
%t A211264 {y, x + 1, b}]]
%t A211264 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A211264 Table[c[n, n], {n, 1, z1}]           (* A056924 *)
%t A211264 Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
%t A211264 Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
%t A211264 Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
%t A211264 Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
%t A211264 Print
%t A211264 c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
%t A211264 Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
%t A211264 Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
%t A211264 Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
%t A211264 Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
%t A211264 Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)
%o A211264 (Magma) [0] cat [&+[(&+[p[2]: p in Factorization(i)] mod 2) *Floor(n div i):i in [2..n] ]:n in [2..65]]; // _Marius A. Burtea_, Oct 17 2019
%o A211264 (Python)
%o A211264 from math import isqrt
%o A211264 def A211264(n): return (lambda m: sum(n//k for k in range(1, m+1))-m*(m+1)//2)(isqrt(n)) # _Chai Wah Wu_, Oct 08 2021
%Y A211264 Cf. A211266, A001222, A006218, A000196.
%Y A211264 Cf. A066829, A094820, A181972.
%K A211264 nonn
%O A211264 1,3
%A A211264 _Clark Kimberling_, Apr 06 2012