|
%I #17 Jul 25 2017 02:33:51
%S 0,1,1,1,1,1,3,3,1,2,3,5,2,3,6,15,1,6,8,3,15,9,4,65,44,82,83,77,75,48,
%T 26,43,1
%N Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.
%C For transient lengths of iterations A062401(x) or A062402(x), if started at 2^n, holds that A096994(n)+1 = a(n). Corresponding cycle lengths satisfy A096852(n-1) = A096857(n). Behind these observation several relationships stand, e.g., sigma(A062401(x)) = A062402(sigma(x)) or phi(A062402(x)) = A062401(phi(x)).
%C For initial value = 2^33 more than 38000 iterations did not lead to a recurrent term, so possibly there is no cycle. a(34) through a(39) are 8, 52, 71, 24, 40, 12. - _Klaus Brockhaus_, Jul 19 2007
%e Trajectory of 2^0 is 1,1, ...; there are zero transient terms preceding the 1-cycle (1), so a(0) = 0.
%e Trajectory of 2^14 is 16384, 16383, 34200, 30480, 26520, 16380, 10200, 6138, 6045, 9906, 9920, 12264, 10200, ...; there are six transient terms preceding the 6-cycle (10200, 6138, 6045, 9906, 9920, 12264), so a(14) = 6.
%t With[{nn = 10^4}, Table[Count[Values@ PositionIndex@ NestList[DivisorSigma[1, EulerPhi@ #] &, 2^n, nn], _?(Length@ # == 1 &)], {n, 0, 60}] /. m_ /; m == nn + 1 -> -1] (* _Michael De Vlieger_, Jul 24 2017 *)
%Y Cf. A062402, A062401, A096857, A096852, A096994.
%K nonn,more
%O 0,7
%A _Labos Elemer_, Jul 22 2004
%E Edited and corrected by _Klaus Brockhaus_, Jul 19 2007
|
