A096994 - OEIS

%I #26 Dec 12 2021 20:09:05

%S 0,0,0,0,0,2,2,0,1,2,4,1,2,5,14,0,5,7,2,14,8,3,64,43,81,82,76,74,47,

%T 25,42,0

%N Number of transient terms if f(x)=phi(sigma(x))=A062401 is iterated at initial value 2^n. Equilibrium terms are listed in A096852.

%C For transient lengths for iterations of A062401(x) or A062402(x) if started at 2^n, A096994(n) + 1 = A096995(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observations several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).

%e n=0: trajectory = {1,1,..} so a(0)=0;

%e n=14: transient-length=14, cycle-length=2, a(14)=14, A096852(14)=2; trajectory ={16384, 27000, 23040, 21600, 17280, 15360, 15488, 13824, 9600, 7680, 7200, 12960, 11880, 11520, [10368,14080], 10368, ...}.

%e Values of a(n) for n > 31, with -1 signifying transient lengths yet unknown after 10^4 iterations of f(x): -1, 7, 51, 70, 23, 39, 11, -1, 37, 107, 30, -1, 145, 25, 21, 36, -1, -1, -1, -1, 31, -1, 452, -1, 449, 447, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 40, -1, -1, -1, -1, -1, -1, -1, 279, -1. - _Michael De Vlieger_, May 15 2017

%t With[{nn = 10^3}, Table[Count[Values@ PositionIndex@ #, k_ /; Length@ k == 1] &@ NestList[EulerPhi@ DivisorSigma[1, #] &, 2^n, nn] /. k_ /; k == nn + 1 -> -1, {n, 31}] ] (* _Michael De Vlieger_, May 15 2017, Version 10 *)

%Y Cf. A096852, A062401.

%K nonn,more

%O 0,6

%A _Labos Elemer_, Jul 22 2004