A352545 - OEIS

%I #13 Mar 30 2022 07:38:28

%S 1611,18523,207441,305429

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

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

%C 3886083 is also in the sequence, cf. EXAMPLE.

%C a(5) > 500000.

%C 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.

%e 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.

%e 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.

%e 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.

%e 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.

%e 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.

%e 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).

%o (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])}

%o a=[0];forstep(n=1,1e5,2,my(t=check(n)); t && #a<#(a=setunion(a,[t])) && print(a[^1]" at n = "n))

%Y Cf. A352544 and references there.

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

%K nonn,base,hard,more

%O 1,1

%A _M. F. Hasler_, Mar 20 2022