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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 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 (ie. 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} signum(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 * signum(a(i+1) - a(i)).

LINKS

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

FORMULA

a(n) = Prod_{i=1..n} i^(1 - ceil(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

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)