A081885 Number of steps needed to reach an integer starting with n+1/16 and iterating the map x-->x*ceiling(x).

8, 5, 3, 8, 18, 8, 5, 6, 5, 6, 3, 6, 7, 5, 1, 6, 4, 6, 3, 4, 2, 6, 2, 7, 3, 2, 4, 6, 2, 5, 1, 8, 12, 4, 7, 4, 8, 7, 3, 9, 10, 5, 3, 9, 5, 6, 1, 3, 11, 4, 2, 4, 4, 4, 2, 4, 6, 7, 2, 2, 5, 3, 1, 7, 9, 9, 5, 7, 4, 13, 4, 5, 6, 8, 6, 14, 8, 3, 1, 4, 2, 5, 4, 5, 3, 2, 2, 10, 2, 5, 3, 18, 4, 3, 1, 11, 4, 7, 7, 4, 5

Offset: 1

Author

Benoit Cloitre, Apr 13 2003

Keywords

• nonn

Comments

Conjecture : let b(n,m) denotes the number of steps needed to reach an integer starting with n+1/2^m and iterating the map x-->x*ceiling(x); then sum(k=1,n,b(k,m)) is asymptotic to (m+1)*n.

Crossrefs

Cf. A068119, A073524.

Programs

Mathematica

Table[Length[NestWhileList[# Ceiling[#]&,n+1/16,!IntegerQ[#]&]]-1, {n,110}] (* Harvey P. Dale, Apr 02 2012 *)

PARI

a(n)=if(n<0,0,s=n+1/16; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Formula

It seems that sum(k=1, n, a(k)) is asymptotic to 5n.