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

1,2

Comments

In these iteration chains, powers of 2 seem to be in the minority.

The sequence is not monotonic.

Links

Table of n, a(n) for n=1..46.

Example

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.

Mathematica

a[n_] := Max@ IntegerExponent[ NestWhileList[# - EulerPhi[#] &, Times @@ Prime[Range[n]], # > 1 &], 2]; Array[a, 25] (* Giovanni Resta, May 30 2018 *)

Prog

(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

Crossrefs

Cf. A002110, A051953.

Sequence in context: A121380 A019759 A019965 * A295798 A072728 A158185

Adjacent sequences: A053145 A053146 A053147 * A053149 A053150 A053151

Keyword

more,nonn

Author

Labos Elemer, Feb 28 2000

Extensions

More terms from R. J. Mathar, May 19 2006

a(37)-a(46) from Giovanni Resta, May 31 2018

Status

approved