A352543 Numbers that end in a loop of size > 2 under iterations of A352544 (= half or add largest anagram).

549, 639, 801, 1035, 1098, 1278, 1503, 1602, 1611, 2025, 2070, 2196, 2511, 2556, 3006, 3123, 3159, 3204, 3222, 3411, 3861, 4050, 4140, 4149, 4383, 4392, 4635, 5022, 5112, 5589, 5679, 5913, 6012, 6165, 6246, 6318, 6345, 6408, 6444, 6795, 6813, 6822, 7047, 7245, 7713, 7722, 7785, 8100, 8280, 8298, 8757, 8766, 8784, 9203, 9270, 9459

Offset: 1

Author

M. F. Hasler, Mar 20 2022

Keywords

• nonn
• base

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.

Examples

			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.
		

Crossrefs

Cf. A352544 (the iterated map) and further references there: A352540 (starting values not ending in a loop), A352541 (number of iterations to reach a value for the second time), A352545 (representatives of cycles of length > 2).

Programs

PARI

is_A352543(n,L=99,a=[n])={for(i=1,L, a=concat(a,n=A352544(n)); #Set(a)>i||break); #a < L && #Set(a[-3..-1]) > 2}