A074735 Number of steps to reach an integer starting with (n+3)/4 and iterating the map x -> x*ceiling(x).

0, 3, 1, 2, 0, 3, 2, 8, 0, 1, 1, 1, 0, 3, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 0, 7, 4, 4, 0, 4, 1, 2, 0, 4, 2, 3, 0, 1, 1, 1, 0, 2, 3, 4, 0, 2, 1, 8, 0, 4, 2, 3, 0, 1, 1, 1, 0, 6, 5, 4, 0, 3, 1, 2, 0, 5, 2, 4, 0, 1, 1, 1, 0, 5, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 0, 4, 4, 5, 0, 6, 1, 2, 0, 3, 2, 5, 0

Offset: 1

Author

Benoit Cloitre, Sep 05 2002

Keywords

• nonn

Comments

Let S(n) = Sum_{k=1..n} a(k) then it seems that S(n) is asymptotic to 2n. S(n)=2n for many values of n, namely n=10,128,198,199,237,238,241,242,246,247,249,267,329... More generally, starting with (n+2^m-1)/2^m and iterating the same map seems to produce the same kind of behavior for a(n) (i.e., Sum_{k=1..n} a(k) is asymptotic to c(m)*n where c(m) depends on m and c(m) is a power of 2).

Crossrefs

Cf. A073524, A073341, A068119.

Programs

Mathematica

Table[Length[NestWhileList[# Ceiling[#]&,(n+3)/4,!IntegerQ[#]&]]-1,{n,110}] (* Harvey P. Dale, Apr 11 2020 *)

PARI

a(n)=if(n<0,0,s=(n+3)/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Formula

Special cases: for k>= 0 a(4k+1) = 0, a(16k+10) = a(16k+11) = a(16k+12) = 1.

Extensions

Offset corrected by Sean A. Irvine, Jan 25 2025