

A228729


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


0



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,4


COMMENTS

Squares of A214080, n > 0. Also, the nth 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



