A339217 -id:A339217 - cp's OEIS frontend

A339918 a(n) = Sum_{k=1..n} floor(3*n/k).

0, 3, 9, 16, 25, 33, 43, 53, 64, 73, 86, 96, 110, 118, 133, 145, 158, 169, 182, 196, 211, 221, 237, 247, 266, 276, 291, 306, 321, 331, 350, 362, 379, 391, 407, 423, 438, 450, 467, 481, 502, 511, 530, 542, 561, 575, 590, 606, 626, 638, 655, 669, 690, 698, 721

Offset: 0

Vaclav Kotesovec, Dec 23 2020

nonn

In general, for m>=1, Sum_{k=1..n} floor(m*n/k) ~ m*n * (log(m*n) + 2*gamma - H(m)), where H(m) = A001008(m)/A002805(m) is the m-th harmonic number and gamma is the Euler-Mascheroni constant A001620.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006218, A339217, A339919.

Table[Sum[Floor[3*n/k], {k, 1, n}], {n, 0, 100}]

a(n) = sum(k=1, n, floor(3*n/k)); \\ Michel Marcus, Dec 23 2020

a(n) ~ 3*n * (log(3*n) + 2*gamma - 11/6), where gamma is the Euler-Mascheroni constant A001620.

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.

Original entry on oeis.org

0, 1, 3, 6, 10, 13, 18, 23, 27, 32, 39, 42, 50, 55, 60, 67, 74, 79, 87, 92, 99, 106, 115, 118, 128, 135, 140, 149, 158, 161, 172, 179, 187, 194, 201, 208, 219, 226, 233, 240, 252, 255, 268, 273, 280, 293, 300, 305, 316, 325, 333, 340, 353, 356, 367, 376, 385, 394, 403, 408, 424, 429

Offset: 1

Rickey W. Austin, Jul 09 2025

nonn

The count excludes points on the base and edges.

			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.

Cf. A000005 (number of divisors), A032741 (divisors < n), A339217 (for optimized generation of lattice divisor patterns).

a(n) = sum(x=1, 2*n-1, sumdiv(x, y, y < min(x, 2*n-x))); \\ Michel Marcus, Jul 11 2025

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))
print([a(n) for n in range(1, 63)])

a(n) = |{ (x, y) : 1 <= x <= 2n - 1, 1 <= y < min(x, 2n - x), and y divides x }|.

cp's OEIS Frontend

A339918 a(n) = Sum_{k=1..n} floor(3*n/k).

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