OFFSET
1,1
COMMENTS
Start by removing the Greedy Cantor's Dust Partition (A348636) from the set of positive integers. We are left with: 3,6,7,8,9,12,15,16,17,... Now apply the A348636 algorithm on this set. Starting with the lowest term, consecutively partition the positive integers excluding the elements of Cantor's set (level 2) into sets s(1), s(2), s(3), ... so that no arithmetic progression of length 3 exists in a set. When choosing s(k), always choose k as small as possible. a(n) = smallest number in s(n).
This process can be applied iteratively to create an infinite sequence of Cantor-Dust-like sequences whose intersection is empty and whose union is the set of positive integers.
Define the original Cantor Dust sequence to be CD(1) and this CD(2). Is it always true for n < m that the number of elements less than k that are in CD(n) will always be equal to or more than the number of elements in CD(m)?
LINKS
Gordon Hamilton, Mini Mathematical Universe - The Jumping Hare, MathPickle, 2022, for teachers of grades 3-6.
EXAMPLE
a(1) = 3, the first integer not in Greedy Cantor's Dust Partition (A348636).
a(2) = 6, the second integer not in Greedy Cantor's Dust Partition.
a(3) = 7, the third integer not in Greedy Cantor's Dust Partition and also not forming an arithmetic progression of length 3 with a(1) and a(2).
a(4) is not 8 because a(2), a(3), 8 would form an arithmetic progression of length 3.
a(4) is not 9 because a(1), a(2), 9 would form an arithmetic progression of length 3.
a(4) = 12, the sixth integer not in Greedy Cantor's Dust Partition and also not forming an arithmetic progression of length 3 with previously admitted integers.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gordon Hamilton, Mar 31 2024
STATUS
approved