A355240 -id:A355240 - 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

A355514 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that q - j/k is a new minimum, i.e., q is approximated from below.

Original entry on oeis.org

1, 3, 8, 13, 44, 75, 106, 243, 380, 517, 654, 791, 2510, 4229, 5948, 7667, 9386, 11105, 12824, 14543, 16262, 17981, 19700, 21419, 23138, 24857, 26576, 28295, 30014, 31733, 33452, 35171, 36890, 38609, 40328, 122703, 205078, 492531, 27869189, 166722603, 305576017

Offset: 1

Hugo Pfoertner, Jul 05 2022

nonn

Cf. A102525, A355512, A355513, A355515.

Terms are candidates for being in A355240, which shares 3, 8, 13, 44, 75.

a355514(upto) = {my(q=log(2)/log(3), dmin=oo);for (m=1, upto, my(n=floor(m*q), qq=n/m, d=q-qq); if (d

A355568 Numbers k > 4 in a Collatz trajectory reaching k after starting at k-1.

Original entry on oeis.org

8, 10, 16, 20, 26, 34, 40, 52, 92, 122, 160, 167, 184, 244, 251, 334, 377, 412, 433, 488, 502, 650, 668, 866, 890, 976, 1154, 1186, 1300, 1336, 1732, 1780, 2308, 3644, 4858, 7288

Offset: 1

Hugo Pfoertner, Jul 10 2022

			8 is a term because the orbit started at 8 - 1 = 7 reaches 8:
  7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8;
10 is a term because it is in the orbit starting at 10 - 1 = 9:
  9 -> 28 -> 14 -> 7 -> 22 -> 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, A355569.

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) = A070993(n+1) + 1.

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.

cp's OEIS Frontend

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

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A355514 Sum of numerator and denominator in a rational approximation j/k of q = log(2)/log(3), such that q - j/k is a new minimum, i.e., q is approximated from below.

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A355568 Numbers k > 4 in a Collatz trajectory reaching k after starting at 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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