A342842 All positive integer solutions m of equation A342369^k(6*p - 2) = m*3 + 2, sorted by p and k in ascending order, p has higher priority than k. p and k are positive integers. "^k" means recursion here.

2, 1, 6, 8, 5, 3, 4, 10, 14, 9, 12, 16, 18, 24, 32, 21, 28, 22, 26, 17, 11, 7, 30, 40, 34, 38, 25, 42, 56, 37, 46, 50, 33, 44, 29, 19, 54, 72, 96, 128, 85, 58, 62, 41, 27, 36, 48, 64, 66, 88, 70, 74, 49, 78, 104, 69, 92, 61, 82, 86, 57, 76, 90, 120, 160, 94, 98, 65, 43

Offset: 1

Author

Thomas Scheuerle, Mar 24 2021

Keywords

• nonn

Comments

It is conjectured that this sequence is a permutation of the positive integers. If it does not contain all positive integers, then there exists a number of the form q = p*6 - 2, where no solution for j*3 - 1 = A006370^k(q) can be found for any j and any k. Such an example is not yet known.

If the sequence were to contain a positive integer more than once, this would mean that A340407 contains a term of uncountable size, which is not the case.

Let us assume here that this sequence is a permutation, then let a'(m) be the inverse permutation, such that a'(a(n)) = n.

Let p = A006370^k(6*(a(n) + 1) - 2) and choose k such that p is of the form m*6 + 4, then a'((p + 2)/6 - 1) < n.

Infinitely many formulas can be developed from this template: a(Sum_{k=1..3^d*n - b} A340407(k) + c) = e*n - f. c is here in the range 0 to d-1 if d-1 > 0. b can be any element of row d in A342261. For all combinations of d, b and c we may find a suitable e and f.

Links

• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A342369, A340407, A006370, A342261.

Programs

MATLAB

function a = A342842( max_p )
    c = 1;
    for p = 1:max_p
        s = 6*p -2;
        while mod(s,3) ~= 0
            s = A342369( s );
            if mod(s,3) == 2
                a(c) = (s-2)/3;
                c = c+1;
            end
        end
    end
end
function b = A342369( n )
    if mod(n,3) == 2
        b = (2*n - 1)/3;
    else
        b = 2*n;
    end
end

Formula

a(1 + Sum_{k=1..n-1} A340407(k)) = 4*n-2.

a(Sum_{k=1..9*n-8} A340407(k)) = 24*n-23.

a(Sum_{k=1..9*n-1} A340407(k)) = 48*n-8.

a(n) = 8*(10^m - 1)/3 + 1 if n = Sum_{k=1..10^m} A340407(k).

a(n) = 4*10^m - 2 if n = -1 + Sum_{k=1..10^m} A340407(k).

a(n) = 4*10^m - 6 if n = -2 + Sum_{k=1..10^m} A340407(k).

a(n) = 5*10^m + (10^(n - 1) - 1)/3 - 13

if n = -3 + Sum_{k=1..10^m} A340407(k).

a(n) = 4*10^m - 10 if n = -4 + Sum_{k=1..10^m} A340407(k).