A216762 a(n) = n * ceiling(log_2(n)) * ceiling(log_2(log_2(n))) * ceiling(log_2(log_2(log_2(n)))) ....

1, 2, 6, 8, 30, 36, 42, 48, 72, 80, 88, 96, 104, 112, 120, 128, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 1188, 1224, 1260, 1296, 1332, 1368, 1404, 1440, 1476, 1512, 1548, 1584, 1620, 1656, 1692, 1728, 1764, 1800

Offset: 1

Author

Ken Takusagawa, Sep 15 2012

Keywords

• nonn

Comments

a(n) is the product of n, ceiling(log_2(n)), ceiling(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 * ceiling(log_2(15)) * ceiling(log_2(log_2(15))) * ceiling(log_2(log_2(log_2(15)))) = 15 * 4 * 2 * 1 = 120.
a(17) = 17 * ceiling(log_2(17)) * ceiling(log_2(log_2(17))) * ceiling(log_2(log_2(log_2(17)))) * ceiling(log_2(log_2(log_2(log_2(17))))) = 17 * 5 * 3 * 2 * 1 = 510.
		

Crossrefs

Cf. A216761 (floor instead of ceiling).

Programs

Haskell

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

Mathematica

Table[prod = 1; s = n; While[s > 1, prod = prod*Ceiling[s]; s = Log[2, s]]; prod, {n, 50}] (* T. D. Noe, Sep 24 2012 *)

PARI

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