My friend John asked me what is the next number in the sequence $$1,4,8,16,17,44,58,76,\dots ?$$ I told him it could be anything, and I asked for a clue. He told me it involves adding numbers that have no prime factor in it more than once like $4$ or $12$ because they have a factor of the prime $2$ twice.
I’m asking for possible solutions to this puzzle not for a definite answer. I tried to look this series up on the oeis but it didn’t show up there.
I think I have an idea for a possible way he got the sequence. he could have added rows of this triangle
1
2,2
3,2,3
2,6,6,2
5,2,3,2,5
…
where its the triangle of the multiplication table but if it has an extra prime factor than it’s prime factor is taken off.
1
2,2
3,4,3
4,6,6,4
5,8,9,8,5
…