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Number of integer pairs (x,y) such that 0<x<=y<=n and x*y=3n.
8

%I #10 Jan 15 2025 22:45:39

%S 0,0,1,1,1,1,1,2,1,2,1,3,1,2,2,3,1,2,1,4,2,2,1,4,2,2,2,4,1,4,1,4,2,2,

%T 3,4,1,2,2,6,1,4,1,4,3,2,1,6,2,4,2,4,1,3,3,6,2,2,1,7,1,2,3,5,3,4,1,4,

%U 2,6,1,6,1,2,4,4,3,4,1,8,2,2,1,7,3,2,2,6,1,6,3,4,2,2,3,7,1,4,3,7,1,4,1,6,5,2,1,6

%N Number of integer pairs (x,y) such that 0<x<=y<=n and x*y=3n.

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

%H Antti Karttunen, <a href="/A211271/b211271.txt">Table of n, a(n) for n = 1..20000</a>

%e a(3) counts this pair: (3,3). - _Antti Karttunen_, Jan 15 2025

%e a(20) counts these pairs: (3,20), (4,15), (5,12), (6,10).

%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 (PARI) A211271(n) = { my(n3=3*n); sumdiv(n3,d,(d <= (n3/d) && (n3/d) <= n)); }; \\ _Antti Karttunen_, Jan 15 2025

%Y Cf. A211266.

%Y Cf. also A211262.

%K nonn

%O 1,8

%A _Clark Kimberling_, Apr 07 2012

%E Data section extended up to a(108) and a(3) corrected from 0 to 1 by _Antti Karttunen_, Jan 15 2025