A053045 a(n) is the number of powers of 2 among the iterates of the Euler phi function when it is iterated with initial value n!.

1, 2, 2, 4, 6, 7, 8, 11, 11, 14, 17, 19, 21, 23, 25, 29, 33, 34, 35, 39, 40, 44, 48, 51, 55, 58, 58, 61, 64, 67, 70, 75, 78, 83, 86, 88, 90, 92, 94, 99, 104, 106, 108, 113, 115, 120, 125, 129, 131, 136, 140, 144, 148, 149, 154, 158, 159, 163, 167, 171, 175, 179, 180

Offset: 1

Author

Labos Elemer, Feb 25 2000

Keywords

• nonn

Comments

Powers of 2 arise at the end of iterations without interruption. Analogous to A053035.

Examples

			For n = 10, the initial value is 10! = 3628800 and the iteration chain is {3628800, 829440, 221184, 73728, 24576, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}. Its length is 19 and 14 values are powers of 2: 8192, ..., 1. Thus a(10) = 14.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Michael De Vlieger)

Crossrefs

Cf. A000010, A000142, A049113, A053035.

Programs

Maple

A053045 := proc(n)
    local a,e;
    e := n! ;
    a :=0 ;
    while e > 1 do
        if isA000079(e) then
            a := a+1 ;
        end if;
        e := numtheory[phi](e) ;
    end do:
    1+a;
end proc:
seq(A053045(n),n=1..18) ; # R. J. Mathar, Jan 09 2017

Mathematica

Table[Count[NestWhileList[EulerPhi, n!, # > 1 &], ?(IntegerQ@ Log2@ # &)], {n, 63}] (* _Michael De Vlieger, Aug 15 2017 *)

Formula

a(n) = A049113(n!). - R. J. Mathar, Jan 09 2017