A216761 n * floor(log_2(n)) * floor(log_2(log_2(n))) * floor(log_2(log_2(log_2(n)))) ....

1, 2, 3, 8, 10, 12, 14, 24, 27, 30, 33, 36, 39, 42, 45, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 320, 330, 340, 350, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 550

Offset: 1

Author

Ken Takusagawa, Sep 15 2012

Keywords

• nonn

Comments

a(n) is the product of n, floor(log_2 n), floor (log_2(log_2 n)), ... with the base-2 logs iterated while the result remains greater than unity.

The sum of the reciprocals of a(n) diverge, but extremely slowly.

In particular, the sum of the reciprocals acts like lg* n asymptotically, where lg* x = 0 for x < 2 and lg* 2^x = 1 + lg* x. - Charles R Greathouse IV, Sep 25 2012

Examples

			a(0) is the product of 0 numbers, defined to be 1.
a(15) = 15 * floor(log_2 15) * floor(log_2 log_2 15) = 15 * 3 * 1 = 45.
a(17) = 17 * floor(log_2 17) * floor(log_2 log_2 17) * floor(log_2 log_2 log_2 17) = 17 * 4 * 2 * 1 = 136.
		

Crossrefs

Cf. A216762 (ceiling instead of floor).

Programs

Haskell

a = product . map floor . takeWhile (1<) . iterate log_2

Mathematica

Table[prod = 1; s = n; While[s > 1, prod = prod*Floor[s]; s = Log[2, s]]; prod, {n, 60}] (* T. D. Noe, Sep 24 2012 *)

PARI

a(n)=my(t=n);n+=1e-9;while(n>2,t*=floor(n=log(n)/log(2)));t \\ Charles R Greathouse IV, Sep 25 2012