A352543 - OEIS

%I #22 Mar 30 2022 06:16:27

%S 549,639,801,1035,1098,1278,1503,1602,1611,2025,2070,2196,2511,2556,

%T 3006,3123,3159,3204,3222,3411,3861,4050,4140,4149,4383,4392,4635,

%U 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

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

%C 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).

%C 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.)

%C All terms below a(54) = 9203 lead to the same loop 1611 -> 7722 -> 3861 -> 12492 -> 6246 -> 3123 -> 6444 -> 3222 -> 1611 of size 8.

%C The starting value 9203 is the only one below 10^4 leading to a different loop, of size 22: cf. EXAMPLE.

%C Sequence A352545 lists the representatives (smallest elements) of the distinct cycles of length > 2.

%e 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.

%e 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.

%o (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}

%Y 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).

%K nonn,base

%O 1,1

%A _M. F. Hasler_, Mar 20 2022