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Exponent of largest power of 2 which appears in the cototient-iteration started with n!.
3

%I #17 Aug 17 2017 22:39:41

%S 0,1,2,4,5,7,9,12,14,11,13,16,14,16,23,27,23,23,24,21,23,28,42,46,26,

%T 26,26,36,43,29,50,55,37,37,40,40,39,59,39,44,68,42,42,44,51,45,50,53,

%U 49,52,51,85,55,57,53,57,60,85,62,71,62,63,60,66,66,107,67,101,76,70,75,77

%N Exponent of largest power of 2 which appears in the cototient-iteration started with n!.

%C If the exponent is a(n), then the number of powers of 2 in the iteration-chain is 1+a(n), the maximal 2-power is 2^a(n) and the number of iterations (until fixed state) performed on these 2-powers is a(n).

%e For n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and starting the iteration of A051593 with n!, the first powers of 2 which appear are 1, 2, 4, 16, 32, 128, 512, 4096, 16384, 2048 and the corresponding exponents are a(n) = 0, 1, 2, 4, 5, 7, 9, 12, 14, 11.

%t Log2 /@ Table[NestWhile[# - EulerPhi@ # &, n!, ! IntegerQ@ Log2@ # &], {n, 60}] (* _Michael De Vlieger_, Aug 15 2017 *)

%o (PARI) cototient(x)= x - eulerphi(x)

%o FunctionIterate(f,x,t)= {local(retval); retval = vector(0); while(x!=t, x = eval(concat(f,"(x)")); retval = concat(retval,x)); retval;}

%o A053039(x) = {local(li,fa,retval); count = 0; li = concat([x! ], FunctionIterate("cototient", x!, 0)); for(i=1,#li, fa = factor(li[i]); if(((matsize(fa)[1] == 1) && (fa[1,1] == 2)),retval = fa[1,2]; break)); retval}

%o for(i=1,72,print1(A053039(i),", ")) \\ _Olaf Voß_, Feb 21 2008

%Y Cf. A051953, A053038, A053475.

%K nonn

%O 1,3

%A _Labos Elemer_, Feb 24 2000

%E More terms from _Olaf Voß_, Feb 21 2008