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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!.

%I A053045 #25 Aug 17 2024 09:01:31
%S A053045 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 A053045 58,58,61,64,67,70,75,78,83,86,88,90,92,94,99,104,106,108,113,115,120,
%U A053045 125,129,131,136,140,144,148,149,154,158,159,163,167,171,175,179,180
%N 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!.
%C A053045 Powers of 2 arise at the end of iterations without interruption. Analogous to A053035.
%H A053045 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 A053045 a(n) = A049113(n!). - _R. J. Mathar_, Jan 09 2017
%e A053045 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 A053045 := proc(n)
%p A053045     local a,e;
%p A053045     e := n! ;
%p A053045     a :=0 ;
%p A053045     while e > 1 do
%p A053045         if isA000079(e) then
%p A053045             a := a+1 ;
%p A053045         end if;
%p A053045         e := numtheory[phi](e) ;
%p A053045     end do:
%p A053045     1+a;
%p A053045 end proc:
%p A053045 seq(A053045(n),n=1..18) ; # _R. J. Mathar_, Jan 09 2017
%t A053045 Table[Count[NestWhileList[EulerPhi, n!, # > 1 &], _?(IntegerQ@ Log2@ # &)], {n, 63}] (* _Michael De Vlieger_, Aug 15 2017 *)
%Y A053045 Cf. A000010, A000142, A049113, A053035.
%K A053045 nonn
%O A053045 1,2
%A A053045 _Labos Elemer_, Feb 25 2000