A117620 Start with x=4/3; repeatedly apply the map x -> (x^2) ceiling(x); sequence gives numerators of the resulting sequence of fractions.

4, 32, 4096, 285212672, 3536203627938199896064, 27735467127437590594631628902073909856749798039036448735232

Offset: 1

Author

Jonathan Vos Post, Apr 07 2006

Keywords

• easy
• frac
• nonn

Comments

In this approximate cubing, does an iteration eventually yield an integer, after which denominators are 1? Fractions are 4/3, 32/9, 4096/81, 285212672/2187, 3536203627938199896064/1594323, 27735467127437590594631628902073909856749798039036448735232/2541865828329, 8393707510592229745861012598171776416393703955772365464679357805492895042198412632866136478758067686243059846017657263750451410617880163800261945260539460460740608/6461081889226673298932241.

a(9) has 1343 digits, and is too large for a b-file. - Robert Israel, Jun 15 2016

Examples

			a(4) = 285212672 because (4096/81)^2 * ceiling(4096/81) = (4096/81)^2 * ceiling(4096/81) = * ceiling(50.5679012) = (16777216/6561) * 51 = 285212672/2187.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..8
• J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A072340, A085276, A117596.

Programs

Maple

x[1]:= 4/3:
for n from 1 to 9 do x[n+1]:= x[n]^2*ceil(x[n]) od:
seq(numer(x[i]),i=1..10); # Robert Israel, Jun 15 2016

Extensions

Erroneous term removed by Giovanni Resta, Jun 15 2016