A228729 - OEIS

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

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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