A072340 Number of steps to reach an integer starting with n/3 and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

0, 2, 6, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 1, 1, 0, 2, 4, 0, 3, 4, 0, 1, 1, 0, 22, 7, 0, 2, 5, 0, 1, 1, 0, 7, 2, 0, 4, 2, 0, 1, 1, 0, 2, 5, 0, 13, 9, 0, 1, 1, 0, 3, 3, 0, 2, 3, 0, 1, 1, 0, 3, 2, 0, 5, 2, 0, 1, 1, 0, 2, 3, 0, 8, 3, 0, 1, 1, 0, 5, 4, 0, 2, 4, 0, 1, 1, 0, 14, 2, 0, 3, 2, 0, 1, 1, 0, 2, 9, 0, 3, 9, 0, 1, 1

Offset: 3

Author

N. J. A. Sloane and J. C. Lagarias, Sep 03 2002

Keywords

• nonn

Comments

We conjecture that an integer is always reached.

The occurrence of the first 1, 2, 3, 4 etc. is at the indices 7, 4, 13, 20, 10, 5, 29, 76, 50, 452, 244, 830, 49, 91, 319, 2639, 5753, 2215, 6151, 7148, 280, 28, 1783 - R. J. Mathar, Nov 25 2006

References

• N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Links

• R. J. Mathar, Table of n, a(n) for n = 3..7147
• J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
• N. J. A. Sloane, Seven Staggering Sequences.

Crossrefs

Cf. A085276, A085285, A085286, A073524.

Programs

Maple

g := proc(x) local M,t1,t2,t3; M := 3^100; t1 := ceil(x) mod M; t2 := x*t1; t3 := numer(t2) mod M; t3/denom(t2); end;
f := proc(n) local t1,c; global g; if type(n, 'integer') then RETURN(0); fi; t1 := g(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := g(t1); od; RETURN(c); end;
[seq(f(n/3),n=3..120)]; # this gives the correct answer as long as the answer is < 99.

Mathematica

a[n_] := Module[{x = n/3, s = 0}, While[!IntegerQ[x], x *= Ceiling[x]; s++]; s]; Table[a[n], {n, 3, 107}] (* Jean-François Alcover, Jan 27 2019, from PARI *)

PARI

A072340(n)={ local(x,s) ; x=n/3 ; s=0 ; while( type(x)!="t_INT", x *= ceil(x) ; s++ ; ) ; return(s) ; } { for(n=3,10000, print(n," ",A072340(n)) ; ) ; } \\ R. J. Mathar, Nov 25 2006