login
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.
1
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
1,1
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.
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).
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Thomas Scheuerle, Mar 24 2021
STATUS
approved