cp's OEIS Frontend

A385807 Number of integer lattice points (x, y) strictly inside a triangle of base 2n - 1 and height n - 1, such that 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y | x.

%I A385807 #42 Jul 21 2025 03:05:50
%S A385807 0,1,3,6,10,13,18,23,27,32,39,42,50,55,60,67,74,79,87,92,99,106,115,
%T A385807 118,128,135,140,149,158,161,172,179,187,194,201,208,219,226,233,240,
%U A385807 252,255,268,273,280,293,300,305,316,325,333,340,353,356,367,376,385,394,403,408,424,429
%N A385807 Number of integer lattice points (x, y) strictly inside a triangle of base 2n - 1 and height n - 1, such that 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y | x.
%C A385807 The count excludes points on the base and edges.
%F A385807 a(n) = |{ (x, y) : 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y divides x }|.
%e A385807 For n = 4, the triangle has x in [1,7]. Valid (x, y) points satisfying y < min(x, 8 - x) and y divides x are: (2,1), (3,1), (4,1), (4,2), (5,1). So a(4) = 5.
%o A385807 (Python)
%o A385807 def a(n): return sum(1 for y in range(1, n) for k in range(1, (2*n)//y + 1) if y < min(y*k, 2*n - y*k))
%o A385807 print([a(n) for n in range(1, 63)])
%o A385807 (PARI) a(n) = sum(x=1, 2*n-1, sumdiv(x, y, y < min(x, 2*n-x))); \\ _Michel Marcus_, Jul 11 2025
%Y A385807 Cf. A000005 (number of divisors), A032741 (divisors < n), A339217 (for optimized generation of lattice divisor patterns).
%K A385807 nonn
%O A385807 1,3
%A A385807 _Rickey W. Austin_, Jul 09 2025