Very simple set?

Originally posted by altfell
yup.. you're right.. forgot about that.. 😳

Last chance:
4,15,16,20,24,26,37,40,42,58,61,73,89,145,650

I think that's it altfell, i can't find a number less than 15 that iterates back into the seed set in the required amount of iterations.

If the solution posted by me earlier is correct, than I think is the the one with the fewest numbers. For a set to have fewer than 16 numbers it will have to contain the number that is equal to how many there are. And from what I calculated those numbers don't lead to success. Except for 15.

The smallest solution has 15 numbers, with the smallest possible largest number being 627:
{4, 15, 16, 20, 24, 26, 37, 38, 40, 42, 58, 73, 89, 145, 627}

It is an interesting set

define
f(abcd..) = a^2 + b^2 + c^2 + ..
where abcd.. is the digit representation of a member of Z+

define
nf(abcd..) as the function f operating on abcd.. n times

abcd.. is in the set Sp where p is the smallest n for pf(abcd..) = qf(abcd..) for some q < p

S8 = {2,4,16,20,24,37,40,42,58,61,73,85,89,98,104,145,154,..}
has the interesting property that if abcd.. is in S8 then nf(abcd..) is in S8 for all values of n. this is the only such value of p for which this occurs

Also, for p less than 8 Sp = {}

I'm also pretty sure that for p != 8 if abcd.. is in Sp then f(abcd..) is not in Sp. So it would be equivalent to define the set S8 as abcd.. is in S8 iaoi f(abcd..) is in S8

I'm looking at whether it is possible to define a metric over Z+ as the number of iterations of nf to reach S8 or possibly S8'={16,37,58,89,145,42,20,4} which i think is the smallest subset of S8 to preserve the property f(abcd..) is in S8' if abcd.. is in S8'

it would be an interesting exercise to see if there exists a function g such that Sp = {3,5,7,11,13,..} and ng is in Sp for all values of n ...