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.)

3067, 4088, 4089, 5742, 6135, 7151, 8179, 8263, 8614, 9979, 10904, 10905, 11016, 11017, 11485, 12922, 13304, 13305, 14303, 14538, 14539, 14689, 15303, 15313, 16527, 16891, 17229, 19384, 19385, 19386, 19585, 19959, 20417, 21482, 21791, 21808, 21811, 22035

Offset

1,1

Comments

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.

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Marcus Elia and Amanda Tucker, Consecutive Integers and the Collatz Conjecture, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 15, Paper A54, 2015. (But beware of errors.)

Example

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.

Prog

(SageMath)
def collatz(n) : return 3*n+1 if n%2 else n//2
def isa(n) :
    parityn = paritynp1 = [-1]*3
    valn = n
    valnp1 = n+1
    while valn != valnp1 :
        if valn==1 or valnp1==1 : return False
        parityn = [parityn[1], parityn[2], valn%2]
        paritynp1 = [paritynp1[1], paritynp1[2], valnp1%2]
        valn = collatz(valn)
        valnp1 = collatz(valnp1)
    return [parityn, paritynp1] not in [ [[1, 0, 0], [0, 0, 1]], [[0, 0, 1], [1, 0, 0]] ]

Crossrefs

Cf. A006577, A078417.

Sequence in context: A101806 A038014 A005944 * A274411 A224970 A224116

Adjacent sequences: A274407 A274408 A274409 * A274411 A274412 A274413

Keyword

nonn

Author

Eric M. Schmidt, Jun 21 2016

Status

approved