login
Number of values which are not powers of 2 in the trajectory when A051953 (cototient function) is repeatedly applied starting with n!.
2

%I #11 Aug 17 2017 12:00:50

%S 1,1,2,2,4,5,7,7,9,20,22,23,35,38,35,35,48,54,62,79,79,85,64,65,108,

%T 124,133,130,120,158,128,128,170,181,179,189,220,181,226,228,192,255,

%U 268,268,269,292,291,286,317,324,337,288,354,352,384,378,396,345,426,393

%N Number of values which are not powers of 2 in the trajectory when A051953 (cototient function) is repeatedly applied starting with n!.

%C Unlike the analogous sequence based on A000005, the non-powers 2 which emerge during iteration are initial, consecutive iterates, except the last one=0.

%e n=9, initial value=9!=362880, the successive iterates when the cototient function (A051953) is repeatedly applied are: {362880, 279936, 186624, 124416, 82944, 55296, 36864, 24576, 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0}. This includes 8 initial and 1 terminal (it is the 0) which are not powers of 2. So a(9)=8+1=9. Beside 15 2-powers appear.

%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 A053036(x) = {local(li,fa,count); 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)) || (matsize(fa)[1] == 0),0,count++)); count}

%o for(i=1,64,print1(A053036(i),", ")) \\ _Olaf Voß_, Feb 20 2008

%Y Cf. A051953, A053475.

%K nonn

%O 1,3

%A _Labos Elemer_, Feb 24 2000

%E More terms from _Olaf Voß_, Feb 20 2008