

A268268


a(n) begins the first chain of 7 consecutive positive integers of hvalues 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.


2



943, 1377, 1494, 1495, 1680, 1681, 1682, 1991, 1992, 1993, 2358, 2359, 2987, 2988, 2989, 2990, 2991, 2992, 2993, 2994, 3288, 3289, 3360, 3542, 3543, 3982, 3983, 3984, 3985, 3986, 3987, 3988, 4193, 4481, 4482, 4722, 4723, 4724, 4725, 4897, 4936, 4937, 5313, 5314
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OFFSET

1,1


COMMENTS

Or numbers k such that h(k) + h(k+6) = h(k+1) + h(k+5) and h(k+3) = (h(k) + h(k+6))/2, where h(k) is the length of k, f(k), f(f(k)), ..., 1 in the Collatz (or 3x + 1) problem.
a(1) = A078441(7).
The symmetry can be seen from the differences between consecutive h(k) (see the example).
The 7tuple of consecutive h(k) are symmetric about the central value h(k+3) which are averages of both their immediate neighbors, their second neighbors and their third neighbors.
A majority of numbers of the sequence generate trivial 7tuples {m, m, m, m, m, m, m}.
The 7tuples {h(k)} of the form {m, p, p, p, p, p, q} are generated by the numbers of the sequence 1377, 4897, ...
The 7tuples {h(k)} of the form {m, m, p, m, q, m, m} are generated by the numbers of the sequence 5511, 58757, ...
The 7tuples {h(k)} of the form {m, p, m, m, m, q, p} are generated by the numbers of the sequence 9514, ...
The 7tuples {h(k)} of the form {m, m, p, p, p, q, q} are generated by the numbers of the sequence 21442, 25666, ...
The 7tuples {h(k)} of the form {m, m, m, p, q, q, q} are generated by the numbers of the sequence 55108, 55293, ...


LINKS

Table of n, a(n) for n=1..44.


EXAMPLE

In 7tuple of consecutive {h(k)}: {h(9514),h(9515),h(9516),h(9517),h(9518),h(9519),h(9520)} = {78,52,78,78,78,104,78}, the central value is 78, and 78+78 = 52+104 = 2*78. Hence, 9514 is in the sequence.
Alternatively, the symmetry can be seen from the differences between consecutive {h(k)}. For {78,52,78,78,78,104,78}, the differences {h(k+1)h(k)} are {26,26,0,0,26,26}.


MATHEMATICA

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+6]==f[m+1]+f[m+5]&&f[m+2]+f[m+4]==f[m]+f[m+6]&& f[m]+f[m+6]==f[m+2]+f[m+4]&&f[m+3]==(f[m]+f[m+6])/2, AppendTo[lst, m]], {m, 1, 6000}]; lst


CROSSREFS

Cf. A006577, A078441, A268253.
Sequence in context: A228382 A252295 A252294 * A106431 A234485 A260136
Adjacent sequences: A268265 A268266 A268267 * A268269 A268270 A268271


KEYWORD

nonn


AUTHOR

Michel Lagneau, Jan 29 2016


STATUS

approved



