A355498 a(n) = A000217(A033676(n)) * A000217(A033677(n)).

1, 3, 6, 9, 15, 18, 28, 30, 36, 45, 66, 60, 91, 84, 90, 100, 153, 126, 190, 150, 168, 198, 276, 210, 225, 273, 270, 280, 435, 315, 496, 360, 396, 459, 420, 441, 703, 570, 546, 540, 861, 588, 946, 660, 675, 828, 1128, 756, 784, 825, 918, 910, 1431, 945, 990, 1008, 1140, 1305, 1770

Offset: 1

Author

Steven Lu, Jul 04 2022

Keywords

• nonn

Comments

When n squares are arranged in a rectangular grid which is as nearly square as possible, a(n) represents the count of rectangles in the grid. The whole grid itself must be a rectangle too.

Examples

			For n=10, the grid as nearly square as possible is 2*5. Thus a(10)=3*15=45 is the number of rectangles in this grid.
		

Crossrefs

Cf. A000217, A033676, A033677.

Programs

Mathematica

Table[(# (# + 1)/2) &[
     First[Select[Divisors[n], # >= Sqrt[n] &]]] (# (# + 1)/2) &[
  Last[Select[Divisors[n], # <= Sqrt[n] &]]], {n, 80}]

PARI

t(n) = n*(n+1)/2; \\ A000217
largdiv(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ A033676
a(n) = my(d=largdiv(n)); t(d)*t(n/d); \\ Michel Marcus, Jul 18 2022