A228729 Product of the positive squares less than or equal to n.

1, 1, 1, 4, 4, 4, 4, 4, 36, 36, 36, 36, 36, 36, 36, 576, 576, 576, 576, 576, 576, 576, 576, 576, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 14400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400, 518400

Offset: 1

Author

Wesley Ivan Hurt, Aug 31 2013

Keywords

• nonn
• easy
• less

Comments

Squares of A214080, n > 0. Also, the n-th value of A001044 (The squared factorial numbers) repeated 2n+1 times, n > 0.

The first differences of a(n) are positive when n is a square (i.e., a(n+1) - a(n) > 0) and zero otherwise. This implies that the square characteristic (A010052) can be written in terms of a(n) as A010052(n) = signum(a(n+1) - a(n)), n > 1. Furthermore, the number of squares less than or equal to n is given by Sum_{i=1..n} sign(a(i+1) - a(i)), and the sum of the squares less than or equal to n is given by Sum_{i=2..n} i * sign(a(i+1) - a(i)).

Examples

			a(6) = 4 since there are two squares less than or equal to 6 (1 and 4) and their product is 1*4 = 4.
		

Crossrefs

Cf. A000290, A001044, A010052, A214080.

Programs

Maple

seq(product( (i)^(1 - ceil(sqrt(i)) + floor(sqrt(i))), i = 1..k ), k=1..100);

Mathematica

Table[Times@@(Range[Floor[Sqrt[n]]]^2), {n, 50}] (* Alonso del Arte, Sep 01 2013 *)

Formula

a(n) = Product_{i=1..n} i^(1 - ceiling(frac(sqrt(i)))).

a(n) = A214080(n)^2, n > 0.

Sum_{n>=1} 1/a(n) = BesselI(0, 2) + 2*BesselI(1, 2) - 1. - Amiram Eldar, Aug 15 2025