A053045 - OEIS

%I #25 Aug 17 2024 09:01:31

%S 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,

%T 58,58,61,64,67,70,75,78,83,86,88,90,92,94,99,104,106,108,113,115,120,

%U 125,129,131,136,140,144,148,149,154,158,159,163,167,171,175,179,180

%N 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!.

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

%H Amiram Eldar, <a href="/A053045/b053045.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Michael De Vlieger)

%F a(n) = A049113(n!). - _R. J. Mathar_, Jan 09 2017

%e 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.

%p A053045 := proc(n)

%p     local a,e;

%p     e := n! ;

%p     a :=0 ;

%p     while e > 1 do

%p         if isA000079(e) then

%p             a := a+1 ;

%p         end if;

%p         e := numtheory[phi](e) ;

%p     end do:

%p     1+a;

%p end proc:

%p seq(A053045(n),n=1..18) ; # _R. J. Mathar_, Jan 09 2017

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

%Y Cf. A000010, A000142, A049113, A053035.

%K nonn

%O 1,2

%A _Labos Elemer_, Feb 25 2000