A352543 -id:A352543 - cp's OEIS frontend

A352544 a(n) = n/2 if n is even, otherwise n + A004186(n), where A004186 = arrange digits in decreasing order.

0, 2, 1, 6, 2, 10, 3, 14, 4, 18, 5, 22, 6, 44, 7, 66, 8, 88, 9, 110, 10, 42, 11, 55, 12, 77, 13, 99, 14, 121, 15, 62, 16, 66, 17, 88, 18, 110, 19, 132, 20, 82, 21, 86, 22, 99, 23, 121, 24, 143, 25, 102, 26, 106, 27, 110, 28, 132, 29, 154, 30, 122, 31, 126, 32, 130, 33, 143, 34, 165, 35, 142, 36, 146, 37, 150, 38, 154, 39, 176, 40, 162, 41, 166, 42, 170, 43, 174, 44, 187, 45

Offset: 0

Eric Angelini and M. F. Hasler, Mar 20 2022

A variant of the Collatz (3x+1) map A006370. See A352540 - A352543 for more about iterations of this map.

Eric Angelini, Divide by 2 or add the biggest anagram, math-fun discussion list, Mar 20 2022

Cf. A004186.
Coincides with A064680 (half if even, double if odd) for n < 13.
Cf. A352540 (initial values with infinite orbit under A352544), A352541 (number of iterations of A352544 to reach a value for the second time), A352542 (orbit of 89 under A352544), A352543 (numbers that end in a loop of length > 3 under A352544), A352545 (representatives of loops of length > 3).

apply( {A352544(n)=if(n%2,n+A004186(n),n\2)}, [0..90]) \\ with A004186(n)=fromdigits(vecsort(digits(n),,4))

A352540 Values for which the iteration of A352544 (half if even, add largest anagram if odd) does not end in a loop.

Original entry on oeis.org

89, 109, 117, 137, 149, 178, 187, 203, 205, 207, 209, 213, 217, 218, 223, 225, 234, 239, 247, 253, 255, 257, 267, 273, 274, 277, 279, 293, 295, 297, 298, 299, 307, 319, 327, 335, 347, 356, 365, 374, 405, 406, 407, 409, 410, 414, 415, 418, 426, 427, 434, 436, 437, 445, 446

Offset: 1

M. F. Hasler, Mar 20 2022

The iterated map A352544 is a variant of the Collatz map, A352544(x) = x/2 if x is even, A352544(x) = x + A004186(x) (add x with digits in decreasing order) if x is odd.

All the terms are only conjectured to have this property; we don't have a completely rigorous proof. But for all the listed initial terms, the trajectory quickly reaches numbers with many (>> 10) digits and grows larger with every iteration: When the number is odd and has a digit 0, then its successor is again odd and at least twice as large, most often more than 9 times larger. Roughly 1/10th of the digits are zeros, and similarly for 9s, so as the terms get larger, it becomes increasingly less probable that they could end up having no digit 0 at all, which is only a necessary condition that they might become even and not grow upon for one iteration, but still most likely resume growth immediately after. See sequence A352542, the trajectory of a(1) = 89, for an example studied in detail.

			See A352541 for examples of trajectories which end in a loop, and A352542 for the trajectory of 89 which grows to infinity.

Eric Angelini, Divide by 2 or add the biggest anagram, math-fun discussion list, Mar 20 2022.

Cf. A352544 (iterated map: half if even, add largest anagram if odd), A352541 (number of iterations to see a value again), A352542 (trajectory of 89), A352543 (starting values ending in cycles of length > 2), A352545 (representatives of cycles of length > 2).

select( {is_A352540(n)=!A352541(n)}, [0..500])

{ n >= 0 | A352541(n) = 0 }.

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

Original entry on oeis.org

1611, 18523, 207441, 305429

Offset: 1

M. F. Hasler, Mar 20 2022

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.

			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).

Cf. A352544 and references there.

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

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))

cp's OEIS Frontend

A352544 a(n) = n/2 if n is even, otherwise n + A004186(n), where A004186 = arrange digits in decreasing order.

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A352540 Values for which the iteration of A352544 (half if even, add largest anagram if odd) does not end in a loop.

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

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