A355568 -id:A355568 - cp's OEIS frontend

A355239 Starting values k > 4 of a Collatz iteration reaching either k-1 or k+1.

5, 6, 7, 9, 11, 14, 15, 17, 18, 19, 25, 33, 39, 41, 47, 51, 54, 57, 59, 62, 71, 81, 89, 91, 107, 108, 121, 159, 161, 166, 183, 243, 250, 252, 284, 333, 376, 378, 411, 432, 487, 501, 639, 649, 651, 667, 865, 889, 959, 975, 977, 1153, 1185, 1299, 1335, 1368, 1439, 1731, 1779, 1823, 2159, 2307, 2430, 2735, 3239, 3643, 4103, 4617, 4857, 4859, 6155, 7287, 7289, 9233

Offset: 1

Hugo Pfoertner, Jul 04 2022

nonn

No further terms up to 2*10^9. It is conjectured that this is the full list of starting values of Collatz trajectories reaching k-1 or k+1, and that the number of steps until this happens is one of the 8 terms of A355240.

There are no further terms up to 31100000000. - Dmitry Kamenetsky, Oct 17 2022

Cf. A070991, A070993, A355240, A355568, A355569.

def f(x): return 3*x+1 if x%2 else x//2
def ok(n):
    if n < 5: return False
    ni, targets = n, {1, n-1, n+1}
    while ni not in targets: ni = f(ni)
    return ni in {n-1, n+1}
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Jul 04 2022

A355240 Numbers of steps until the Collatz iteration started at k > 4 returns to either k-1 or k+1.

Original entry on oeis.org

3, 8, 13, 44, 75, 88, 101, 119

Offset: 1

Hugo Pfoertner, Jul 04 2022

It is conjectured that no further terms exist.
Michael S. Branicky checked this up to 2*10^9 and found no starting values other than the 74 terms given in A355239 and also in the attached file.
The terms of this sequence are expected to be close to the sum of numerators and denominators in a rational approximation of log(2)/log(3). See A355514, which shares terms 3, 8, 13, 44, 75.

Hugo Pfoertner, Collatz iterations started at k returning to k+-1.

Cf. A005186, A006577, A355514.
A355239 gives the list of starting values.
Cf. A355568, A355569.

a355240(upto) = {my(D=List()); for (start=5, upto, my(old=start,new=0,L=0);while (abs(new-start)>1 && new!=1, L++; if(old%2==0,new=old/2,new=3*old+1);old=new); if(new>1, listput(~D,L))); Set(D)};
a355240(10000)

A355569 Numbers k > 4 in a Collatz trajectory reaching k after starting at k+1.

Original entry on oeis.org

5, 8, 10, 13, 16, 17, 38, 40, 46, 53, 56, 58, 61, 70, 80, 88, 106, 107, 160, 251, 283, 377, 638, 650, 958, 976, 1367, 1438, 1822, 2158, 2429, 2734, 3238, 4102, 4616, 4858, 6154, 7288, 9232

Offset: 1

Hugo Pfoertner, Jul 10 2022

			a(1) = 5 because the orbit started at 6 = a(1) + 1 reaches 5:
  6 -> 3 -> 10 -> 5;
a(2) = 8 because the orbit started at 9 = a(2) + 1 reaches 8:
  9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8;
a(3) = 10 because the orbit started at 11 = a(3) + 1 reaches 10:
  11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A006577, A070991, A070993, A355239, A355240, A355568.

collatz(start, target) = {my(old=start, new=0); while (new!=target && new!=1, if(old%2==0, new=old/2, new=3*old+1); old=new); new>1};
for (k=5, 10000, if(collatz(k+1, k), print1(k, ", ")))

a(n) = A070991(n+2) - 1.

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A355239 Starting values k > 4 of a Collatz iteration reaching either k-1 or k+1.

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A355240 Numbers of steps until the Collatz iteration started at k > 4 returns to either k-1 or k+1.

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A355569 Numbers k > 4 in a Collatz trajectory reaching k after starting at k+1.

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