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A268486 Least k starting a chain of (2n+1) consecutive integers {h(k+i)}, i=0,1,...,2n, where h(k) is the length of the finite set {k, f(k), f(f(k)), ..., 1} in the Collatz (or 3x + 1) problem, with the property that h(k) = h(k+2n), h(k+1) = h(k+2n-1), ..., h(k+n-1) = h(k+n+1).

%I A268486 #16 Nov 16 2024 21:31:02
%S A268486 24,48,230,229,228,2987,7083,7083,14168,15959,57346,57346,119388,
%T A268486 182852,365740,365739,365738,596310,596310,1088124,1088123,2901713,
%U A268486 2901712,3264428,3264428
%N A268486 Least k starting a chain of (2n+1) consecutive integers {h(k+i)}, i=0,1,...,2n, where h(k) is the length of the finite set {k, f(k), f(f(k)), ..., 1} in the Collatz (or 3x + 1) problem, with the property that h(k) = h(k+2n), h(k+1) = h(k+2n-1), ..., h(k+n-1) = h(k+n+1).
%C A268486 It is interesting to look for symmetries in the sequence A006577 (number of halving and tripling steps to reach 1 in '3x+1' problem). The symmetrical architecture is different from A268468 with h(k) + h(k+2n) = h(k+1)+ h(k+2n-1) = ... = h(k+n-1)+ h(k+n+1) = 2*h(k+n) where h(k+n) is the symmetrical center. Here the property is h(k)= h(k+2n), h(k+1) = h(k+2n-1), ..., h(k+n-1) = h(k+n+1), and the corresponding symmetrical centers h(k+n) have no special properties (for instance, a symmetrical center is not calculable according to the others terms of the chain).
%C A268486 We observe two essential families of chains containing symmetries:
%C A268486 (i) The majority of trivial chains are obtained when a(n) begins the first chain of 2n+1 consecutive positive integers where h(k) = h(k+1) = ... = h(k+2n).
%C A268486 The numbers of the sequence having this property are 2987, 7083, 57346, 596310, 3264428, ...
%C A268486 (ii) Chains having several distinct values. This case is more interesting, with an important question: how many maximum distinct values can contain such a set {h(k)}?
%C A268486 The numbers of the sequence having this property are 24, 48, 230, 229, 228, 14168, 15959, 119388, 182852, 365740, 365739, 365738, 1088124, 2901713, ... For each corresponding chain of the sequence, the number of distinct elements is 2, 2, 4, 4, 4, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 2, 1, 1, 3, 3, 2, 2, 1, ...
%C A268486 The symmetrical centers h(k+n) are 23, 24, 83, 83, 83, 48, 57, 57, 151, 190, 78, 78, 242, 85, 197, ... with the corresponding k = 25, 50, 233, 233, 233, 2993, 7090, 7091, 14177, 15969, 57357, ...
%e A268486 a(3) = 230 because in the first 7-tuple {h(230), h(231), h(232), h(233), h(234), h(235), h(236)} = {34, 127, 21, 83, 21, 127, 34}, the numbers are symmetric w.r.t. its central h(233) = 83. Hence 230 belongs to the sequence.
%e A268486 The first elements k of the other 7-tuples {h(k+i)}, i=0..6, are 362, 810, 836, 943, 1222, 1256, 1322, 1410, ...
%p A268486 nn:=10^7:T:=array(1..nn):
%p A268486 for j from 1 to 5*10^6 do:
%p A268486   k:=0:m:=j:it:=0:
%p A268486     for i from 1 to nn while(m<>1) do:
%p A268486      if irem(m,2)=0
%p A268486       then
%p A268486        m:=m/2:
%p A268486       else
%p A268486       m:=3*m+1:
%p A268486      fi:
%p A268486     it:=it+1:
%p A268486     od:
%p A268486     k:=j:T[j]:=it:
%p A268486    od:
%p A268486    for n from 43 by 2 to 60 do:
%p A268486    ii:=0:
%p A268486    for j from 1 to nn while(ii=0)do:
%p A268486      itr:=0:lst:={}:
%p A268486      for jj from 1 to (n-1)/2 do:
%p A268486       lst:=lst union {T[j+jj-1]} union {T[j+n-jj]}:
%p A268486       if T[j+jj-1]= T[j+n-jj]
%p A268486        then
%p A268486        itr:=itr+1:
%p A268486        else fi:
%p A268486       od:
%p A268486       if itr=(n-1)/2 then ii:=1:
%p A268486       printf("%d %d \n",n,j):
%p A268486       else
%p A268486       fi:
%p A268486      od:
%p A268486      od:
%Y A268486 Cf. A006577, A268468.
%K A268486 nonn,more
%O A268486 1,1
%A A268486 _Michel Lagneau_, Feb 06 2016