A268268 a(n) begins the first chain of 7 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.

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

Offset: 1

Author

Michel Lagneau, Jan 29 2016

Keywords

• nonn

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 7-tuple 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 7-tuples {m, m, m, m, m, m, m}.

The 7-tuples {h(k)} of the form {m, p, p, p, p, p, q} are generated by the numbers of the sequence 1377, 4897, ...

The 7-tuples {h(k)} of the form {m, m, p, m, q, m, m} are generated by the numbers of the sequence 5511, 58757, ...

The 7-tuples {h(k)} of the form {m, p, m, m, m, q, p} are generated by the numbers of the sequence 9514, ...

The 7-tuples {h(k)} of the form {m, m, p, p, p, q, q} are generated by the numbers of the sequence 21442, 25666, ...

The 7-tuples {h(k)} of the form {m, m, m, p, q, q, q} are generated by the numbers of the sequence 55108, 55293, ...

Examples

			In 7-tuple 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}.
		

Crossrefs

Cf. A006577, A078441, A268253.

Programs

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