A049115 - OEIS

%I #33 Aug 28 2021 06:44:59

%S 0,0,1,0,1,1,2,0,2,1,2,1,2,2,1,0,1,2,3,1,2,2,3,1,2,2,3,2,3,1,2,0,2,1,

%T 2,2,3,3,2,1,2,2,3,2,2,3,4,1,3,2,1,2,3,3,2,2,3,3,4,1,2,2,3,0,2,2,3,1,

%U 3,2,3,2,3,3,2,3,2,2,3,1,4,2,3,2,1,3,3,2,3,2,3,3,2,4,3,1,2,3,2,2,3,1,2,2,2

%N a(n) is the number of iterations of the Euler phi function needed to reach a power of 2, when starting from n.

%C a(n) = A227944(n) if n is not a power of 2. - _Eric M. Schmidt_, Oct 13 2013

%H T. D. Noe, <a href="/A049115/b049115.txt">Table of n, a(n) for n = 1..10000</a>

%F The smallest x so that Nest[ EulerPhi, n, x ] = 2^w is just achieved.

%F From _Antti Karttunen_, Aug 28 2021: (Start)

%F If A209229(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A000010(n)).

%F a(n) <= A003434(n) and a(n) <= A329697(n) for all n.

%F (End)

%e If n is a power of 2, then a(n)=0 by definition. If n = 59049, then by iterating with phi, we get 59049 -> 39366 -> 13122 -> 4374 -> 1458 -> 486 -> 162 -> 54 -> 18 -> 6 -> 2 -> 1. It took ten steps to reach the first power of 2 (2 in this case), so a(59049) = 10.

%t Table[If[IntegerQ@ Log2@ n, 0, -1 + Length@ NestWhileList[EulerPhi, n, ! IntegerQ@ Log2@ # &]], {n, 105}] (* _Michael De Vlieger_, Aug 01 2017 *)

%o (PARI) A049115(n) = if(!bitand(n,n-1),0,1+A049115(eulerphi(n))); \\ _Antti Karttunen_, Aug 28 2021

%Y Cf. A000010, A003434, A049113, A057716, A209229, A227944.

%Y Cf. also A064097, A064415, A329697.

%K nonn

%O 1,7

%A _Labos Elemer_

%E Definition corrected and simplified, example corrected by _Antti Karttunen_, Aug 28 2021