A352541 -id:A352541 - 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))

A352542 Trajectory of initial value 89 under iterations of the map A352544: half if even, add largest anagram if odd.

Original entry on oeis.org

89, 187, 1058, 529, 1481, 9892, 4946, 2473, 9905, 19855, 118406, 59203, 154523, 708844, 354422, 177211, 949322, 474661, 1241102, 620551, 1275761, 9040972, 4520486, 2260243, 8692463, 18558895, 117444446, 58722223, 146254445, 801698866, 400849433, 1385292733, 11260625954

Offset: 0

M. F. Hasler, Mar 20 2022

89 is the smallest nonnegative integer with an orbit of infinite size under iterations of x -> A352544(x) = {x/2 if x is even, x + A004186(x) if x is odd}. The list of all such numbers is given in A352540, which contains this sequence as subset.
We conjecture that there is a strictly increasing sequence (b(k), k >= 0) = (32, 37, 46, 52, 88, 91, 118, 122, 141, ...) such that all terms a(n) with n >= b(k) have more than k digits 0.
As a consequence, the sequence tends to a 10-adic limit ...27057751007.
Similarly, the number of leading digits 1 appears to grow to infinity; more precisely, a(n) has more than k leading digits 1 for all n > c(k >= 0) = (50, 70, 95, 121, 122, 123, 130, ...).

			The initial term a(0) = 89 and its successor a(1) = 187 are odd, so the number with the same digits in decreasing order, 98 resp. 871, are added to find the successor a(n+1).
Then a(2) = 1058 is even (as are a(5..6), a(10), a(13..14), ...), so the successor is obtained dividing it by two.
a(32) = 11260625954 appears to be the last even term. It appears that from this terms on, all terms have at least one digit 0 and therefore all subsequent terms end in the digit 7.
From a(37) = 11079547822507 on, all terms appear to have at least two digits 0, and therefore all end in the digits ...07.
From a(46) = 11109941625118561459007 on, all terms appear to have at least three digits 0, and therefore all end in the digits ...007.
From a(52) = 1119999530692487035860091007 on, all terms appear to have at least four digits 0, and therefore all end in the digits ...1007.
a(49) = 9999653161399504894770007 ~ 9.999653e24 appears to be the last term to have:
    (i) not more digits than the preceding term,
   (ii) its leading digit different from 1,
  (iii) a successor a(n+1) ~ 1.999965e25 ~ 2*a(n) and a(51) ~ 1.1999964e26 ~ 6*a(50).
For all n >= 51, a(n) has one more digit than a(n-1), and a(n+1) > 9*a(n).

Eric Angelini, Divide by 2 or add the biggest anagram, math-fun discussion list, Mar 20 2022
M. F. Hasler, Number of digits 0 in A352542(0..500), PARI/GP plot, Mar 22 2022
M. F. Hasler, Number of leading digits 1 in A352542(0..500), PARI/GP plot, Mar 22 2022

Cf. A352544 (the iterated map), A352540 (starting values with infinite orbit), A352541 (number of iterations until a value is repeated).

A352542_upto(N)=vector(N+1,i,N=if(i>1,A352544(N),89))

log_10 a(n) ~ n (asymptotical equivalence, as n -> oo).

a(n+1) > 9*a(n) for all n > 50. (Conjectured.)

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 }.

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

Original entry on oeis.org

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

M. F. Hasler, Mar 20 2022

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.

			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.

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).

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}

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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A352542 Trajectory of initial value 89 under iterations of the map A352544: half if even, add largest anagram if odd.

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

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