Get over 50% chance to get N things At each draw, you take 1 of N things (I'll call them coupons) (Uniformly distributed) with replacement and you take note of which ones you have seen so far. Determine the number n such that after n draws you have at least a 50% chance of seeing all different coupons. Essentially I am asking for a formula in terms of and N for n.
This is a problem similar to the coupon collector problem. Thank you in advance!
So far I tried base examples to derive a formula and prove it with PMI with no luck. I reposted this problem after making some edits to the original (which was put on hold for some reason).

Using my idea and finding the first six entries in the sequence: 1,2,5,7,10,13,... and then searching http://oeis.org yielded the result A073593.
There is no convenient formula listed, but according to the references it is hypothesized that it is asymptotic to $n(\log(n)+\frac{1}{e})$
The method suggested used to generate the list is similar to the one I describe above.  Their specific suggestion is more efficient in that they only care about keeping track of the far right column, or equivalently worded, they search for the least $n$ for which $A^nv$ has top entry greater than $\frac{1}{2}$, but instead of calculating in the order $(A(A(\dots(A))\dots)v$ they calculate instead in the order $(A(A(\dots(Av))\dots)$, requiring less memory and computation time.
Specific pseudocode on oeis: a(n)=v=vector(n+1); k=1; v[n]=1.0; while(v[1]<0.5, k++; for(i=1, n, v[i]=v[i]*(n+1-i)/n+v[i+1]*i/n)); k
