A352545 Representatives, i.e., minimal elements of cycles of length > 2 under iterations of A352544 (half or add largest anagram).

1611, 18523, 207441, 305429

Offset: 1

Author

M. F. Hasler, Mar 20 2022

Keywords

• nonn
• base
• hard
• more

Comments

All terms are odd, since the smallest (resp. largest) element of a cycle of A352544 is always odd (resp. even).

3886083 is also in the sequence, cf. EXAMPLE.

a(5) > 500000.

To compute the sequence we look for odd initial values ending in a cycle of more than two elements. This gives terms of the sequence, but we don't know the position of a term until we have scanned all (relevant) initial values up to that number, cf. EXAMPLES.

Examples

			The starting value x = 549 leads to 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 a(1) = 1611.
The starting value x = 9203 leads to the cycle (18523, 103844, 51922, 25961, 122482, 61241, 125452, 62726, 31363, 94694, 47347, 124790, 62395, 158927, 1146448, 573224, 286612, 143306, 71653, 148184, 74092, 37046, 18523) of length = 22 with (smallest) representative a(2) = 18523.
The starting value x = 36037 leads to 112367 -> 875578 -> 437789 -> 1425532 -> 712766 -> 356383 -> 1221716, element of the cycle (610858, 305429, 1259749, 11235170, 5617585, 14383136, 7191568, 3595784, 1797892, 898946, 449473, 1423916, 711958, 355979, 1353532, 676766, 338383, 1221716) of length 18, with (smallest) representative a(4) = 305429.
The starting value x = 84807 leads to 173547 -> ... -> 5637789 -> 15515442 which is part of the cycle (7772121, 15544332, 7772166, 3886083, 12772413, 90204624, 45102312, 22551156, 11275578, 5637789, 15515442, 7757721, 15535242, 7767621, 15544242) of length 15 and (smallest) representative 3886083.
The starting value x = 104481 leads to 948591 -> 1947132 part of the cycle (973566, 486783, 1374426, 687213, 1563534, 781767, 1659528, 829764, 414882, 207441, 951651, 1917162, 958581, 1947132) of length 14 with (smallest) representative a(3) = 207441.
We actually don't know that this is a(3) until we have checked that no smaller starting value will produce a smaller term. Similarly, we know the index of a(4) only after checking all (odd) starting values less than a(4).
		

Crossrefs

Cf. A352544 and references there.

Subsequence of A352543: starting values ending in a loop of size > 2.

Programs

PARI

check(n,L=1e99,U=List(n),i)={ while(i=setsearch(U, n=A352544(n),1), n>L&&return; listinsert(U,n,i)); U=List(n); while(U[1]!=n=A352544(n), listput(U,n)); if(#U>2,Set(U)[1])}
a=[0];forstep(n=1,1e5,2,my(t=check(n)); t && #a<#(a=setunion(a,[t])) && print(a[^1]" at n = "n))