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%I #8 Apr 11 2020 17:24:16
%S 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,
%T 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,
%U 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
%N Number of steps to reach an integer starting with (n+3)/4 and iterating the map x -> x*ceiling(x).
%C 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).
%F Special cases: for k>= 0 a(4k+1) = 0, a(16k+10) = a(16k+11) = a(16k+12) = 1.
%t Table[Length[NestWhileList[# Ceiling[#]&,(n+3)/4,!IntegerQ[#]&]]-1,{n,110}] (* _Harvey P. Dale_, Apr 11 2020 *)
%o (PARI) a(n)=if(n<0,0,s=(n+3)/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)
%Y Cf. A073524, A073341, A068119.
%K nonn
%O 0,2
%A _Benoit Cloitre_, Sep 05 2002
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