A268177 -id:A268177 - cp's OEIS frontend

A268288 a(n) begins the first chain of 9 consecutive positive integers of h-values with symmetrical gaps about the center, where h(k) is the length of the finite sequence k, f(k), f(f(k)), ...., 1 in the Collatz (or 3x + 1) problem.

1680, 1991, 2987, 2988, 2989, 2990, 2991, 2992, 3982, 3983, 3984, 3985, 3986, 4722, 4723, 5313, 5314, 5315, 5316, 5317, 6576, 6577, 6578, 7083, 7084, 7085, 7086, 7087, 7088, 7089, 7090, 7091, 7794, 7795, 7976, 7977, 7978, 7979, 7980, 7981, 8769, 8770, 8771

Offset: 1

Michel Lagneau, Jan 30 2016

nonn

a(1) = A078441(9).
The 9-tuple of consecutive h(k) are symmetric about the central value h(k+4) which are averages of both their immediate neighbors, their second neighbors, their third neighbors and their fourth neighbors.
A majority of numbers the sequence generate trivial 9-tuples (m, m, m, m, m, m, m, m, m).
For a(n) < 200000, the following sets have been identified:
The 9-tuples {h(k)} of the form {m, p, p, p, p, p, p, p, q} are generated by the numbers of the sequence 12608, 16915, 39169, ...
The 9-tuples {h(k)} of the form {m, p, q, q, q, q, q, m, p} are generated by the numbers of the sequence 40553, ...
The 9-tuples {h(k)} of the form {m, p, p, p, q, m, m, m, p} are generated by the numbers of the sequence 55107, 124739, ...
The 9-tuples {h(k)} of the form {m, m, m, m, p, q, q, q, q} are generated by the numbers of the sequence 55292, 90396, 118109, ...
The 9-tuples {h(k)} of the form {m, m, m, p, m, q, m, m, m} are generated by the numbers of the sequence 58756, 71236, 79428, ...
The 9-tuples {h(k)} of the form {m, m, p, m, m, m, q, m, m} are generated by the numbers of the sequence 78021, ...
The 9-tuples {h(k)} of the form {m, p, m, m, m, m, m, q, m} are generated by the numbers of the sequence 93600, 124768, ...
The 9-tuples {h(k)} of the form {m, m, m, p, p, p, q, q, q} are generated by the numbers of the sequence 160705, ...

			In 9-tuple of consecutive h(k): {h(55107),h(55108),...,h(55115)} = {184,60,60,60,122,184,184,184,60}, the central value is 122, and 184+60 = 2*122. Hence, 55107 is in the sequence.
Alternatively, the symmetry can be seen from the differences between consecutive h(k). For {184,60,60,60,122,184,184,184,60}, the differences h(k+1)-h(k) are (-124,0,0,62,62,0,0,-124).

Cf. A006577, A078441, A268177, A268253.

lst={};f[n_]:=Module[{a=n,k=0},While[a!=1,k++;If[EvenQ[a],a=a/2,a=a*3+1]];k];Do[If[f[m]+f[m+8]==f[m+1]+f[m+7]&&f[m+2]+f[m+6]==f[m+3]+f[m+5]&& f[m]+f[m+8]==f[m+3]+f[m+5]&&f[m+4]==(f[m]+f[m+8])/2,AppendTo[lst,m]],{m,1,6000}];lst

cp's OEIS Frontend

A268288 a(n) begins the first chain of 9 consecutive positive integers of h-values with symmetrical gaps about the center, where h(k) is the length of the finite sequence k, f(k), f(f(k)), ...., 1 in the Collatz (or 3x + 1) problem.

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