A solution to this problem?

[Ben1587]

Replace some elements of the natural sequence 1, 2, 3, ... such that after this replacement the sum of first k terms is divisible by k.

Any solutions?

[loseyourname]

Replace every element from 2 onward with 1. Then the sum of the first k terms will be k.

[Muzza]

Replace all elements (except the k:th) with 0. Let the k:th element be k (or ak for some integer a ;)).

[NateTG]

Replace some elements of the natural sequence 1, 2, 3, ... such that after this replacement the sum of first k terms is divisible by k.

Any solutions?

What if by some he means 'a finite subset'?

[Njorl]

Doubling every number works.

sum(1 to k)=k(k+1)/2 so multiplying this by 2 yields k(k+1) which is of course divisible by k.

Still, that is replacing all of the numbers, but I think it is on the road to a more pleasing solution.

Njorl