A053148 When cototient function (A051953) is iterated with initial value A002110(n), a(n) = exponent of the largest power of 2 which appears in the iteration.
1, 2, 3, 5, 5, 8, 5, 9, 8, 16, 6, 9, 12, 8, 9, 7, 4, 11, 6, 6, 9, 13, 8, 13, 11, 17, 7, 13, 20, 4, 11, 11, 15, 13, 9, 19, 13, 6, 4, 13, 4, 4, 8, 4, 24, 20
Offset: 1
Examples
For n=10, the iteration chain of 43 terms is {6469693230, 5447823150, 4315810350, ..., 188416, 98304, 65536, 32768, ..., 4, 2, 1, 0} in which the largest power of 2 is 65536 = 2^16, so a(10)=16;
for n=11 the length is 61, including 54 numbers that are not powers of 2 and 7 powers of 2, of which the largest is 2^6 thus a(11)=6.
Programs
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Mathematica
a[n_] := Max@ IntegerExponent[ NestWhileList[# - EulerPhi[#] &, Times @@ Prime[Range[n]], # > 1 &], 2]; Array[a, 25] (* Giovanni Resta, May 30 2018 *)
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PARI
A051953(n)= { return(n-eulerphi(n)); } A002110(n)= { return(prod(i=1,n,prime(i))); } ispow2(n)= { local(nbin,nbinl,sd); nbin=binary(n); nbinl=matsize(nbin); sd=sum(i=1,nbinl[2],nbin[i]); if(sd==1, return(nbinl[2]-1), return(0); ); } A053148itr(n)= { local(v,vbin,maxp); v=A002110(n); maxp=ispow2(v); while(v>0, v=A051953(v); maxp=max(maxp,ispow2(v)); ); return(maxp); } { for(n=1,70, print1(A053148itr(n),","); ); } \\ R. J. Mathar, May 19 2006
Extensions
More terms from R. J. Mathar, May 19 2006
a(37)-a(46) from Giovanni Resta, May 31 2018
Comments