cp's OEIS Frontend

A274410 Numbers n such that the Collatz iterations for n and n + 1 have the same length (A078417) but do not meet a certain condition. (See comments.)

%I A274410 #17 Mar 07 2020 05:41:02
%S A274410 3067,4088,4089,5742,6135,7151,8179,8263,8614,9979,10904,10905,11016,
%T A274410 11017,11485,12922,13304,13305,14303,14538,14539,14689,15303,15313,
%U A274410 16527,16891,17229,19384,19385,19386,19585,19959,20417,21482,21791,21808,21811,22035
%N A274410 Numbers n such that the Collatz iterations for n and n + 1 have the same length (A078417) but do not meet a certain condition. (See comments.)
%C A274410 Consider the parity vectors of the Collatz iterations of n and n + 1. Consider the portions of the vectors before the Collatz iterations start to coincide. Then the condition to exclude n from the sequence is that these portions end in (0, 0, 1) and (1, 0, 0), in either order.
%H A274410 Eric M. Schmidt, <a href="/A274410/b274410.txt">Table of n, a(n) for n = 1..10000</a>
%H A274410 Marcus Elia and Amanda Tucker, <a href="https://www.emis.de/journals/INTEGERS/papers/p54/p54.Abstract.html">Consecutive Integers and the Collatz Conjecture</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 15, Paper A54, 2015. (But beware of errors.)
%e A274410 The Collatz iterations for 3067 and 3068 yield 1384 on the 27th iteration in both cases. For 3067, the three previous terms are (1844, 922, 461), with parities (0, 0, 1). For 3068, the three previous terms are (11072, 5536, 2768), with parities (0, 0, 0). Thus the condition fails to hold and 3067 is in the sequence.
%o A274410 (Sage)
%o A274410 def collatz(n) : return 3*n+1 if n%2 else n//2
%o A274410 def isa(n) :
%o A274410     parityn = paritynp1 = [-1]*3
%o A274410     valn = n
%o A274410     valnp1 = n+1
%o A274410     while valn != valnp1 :
%o A274410         if valn==1 or valnp1==1 : return False
%o A274410         parityn = [parityn[1], parityn[2], valn%2]
%o A274410         paritynp1 = [paritynp1[1], paritynp1[2], valnp1%2]
%o A274410         valn = collatz(valn)
%o A274410         valnp1 = collatz(valnp1)
%o A274410     return [parityn, paritynp1] not in [ [[1,0,0],[0,0,1]], [[0,0,1],[1,0,0]] ]
%Y A274410 Cf. A006577, A078417.
%K A274410 nonn
%O A274410 1,1
%A A274410 _Eric M. Schmidt_, Jun 21 2016