A096995 Number of transient terms if f(x) = sigma(phi(x)) = A062402 is iterated at initial value = 2^n.

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, 26, 43, 1

Offset

0,7

Comments

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)).

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

Links

Table of n, a(n) for n=0..32.

Example

Trajectory of 2^0 is 1,1, ...; there are zero transient terms preceding the 1-cycle (1), so a(0) = 0.
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.

Mathematica

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 *)

Crossrefs

Cf. A062402, A062401, A096857, A096852, A096994.

Sequence in context: A004550 A096836 A360530 * A255941 A010264 A262816

Adjacent sequences: A096992 A096993 A096994 * A096996 A096997 A096998

Keyword

nonn,more

Author

Labos Elemer, Jul 22 2004

Extensions

Edited and corrected by Klaus Brockhaus, Jul 19 2007

Status

approved