OFFSET
1,1
COMMENTS
Most small starting values end in a cycle or loop of length 2 under iterations of A352544 (like 1 -> 2 -> 1, 3 -> 6 -> 3, or 49 -> 143 -> ... -> 7915 -> 17666 -> 8833 -> 17666, and some (listed in A352540) have an unbounded orbit like 89 (cf. A352542).
This sequence lists all other starting values, i.e., those which have a bounded orbit but don't end in a cycle of length 2. (A352544 obviously has no fixed points.)
All terms below a(54) = 9203 lead to the same loop 1611 -> 7722 -> 3861 -> 12492 -> 6246 -> 3123 -> 6444 -> 3222 -> 1611 of size 8.
The starting value 9203 is the only one below 10^4 leading to a different loop, of size 22: cf. EXAMPLE.
Sequence A352545 lists the representatives (smallest elements) of the distinct cycles of length > 2.
EXAMPLE
The number a(1) = 549 is the smallest starting value which leads into a cycle of length > 2 under iterations of the map A352544: namely, 549 -> 1503 -> 6813 -> 15444 -> 7722 which is element of the cycle [3861, 12492, 6246, 3123, 6444, 3222, 1611, 7722] of length 8, with representative = smallest member A352545(1) = 1611.
The starting value a(54) = 9203 is the only one below 10^4 leading to a different loop: it goes at once to 18523 -> 103844 -> 51922 -> 25961 -> 122482 -> 61241 -> 125452 -> 62726 -> 31363 -> 94694 -> 47347 -> 124790 -> 62395 -> 158927 -> 1146448 -> 573224 -> 286612 -> 143306 -> 71653 -> 148184 -> 74092 -> 37046 -> 18523, a loop of size 22, with representative = smallest member A352545(2) = 18523.
PROG
CROSSREFS
KEYWORD
nonn,base
AUTHOR
M. F. Hasler, Mar 20 2022
STATUS
approved