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

1,4

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)).

Links

Table of n, a(n) for n=1..45.

Formula

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

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

Example

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

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 *)

Crossrefs

Cf. A000290, A001044, A010052, A214080.

Sequence in context: A035627 A228423 A165923 * A174444 A268237 A276866

Adjacent sequences: A228726 A228727 A228728 * A228730 A228731 A228732

Keyword

nonn,easy,less

Author

Wesley Ivan Hurt, Aug 31 2013

Status

approved