Brown's Criterion

A Sequence $\{\nu_i\}$ of nondecreasing Positive Integers is Complete Iff

1. $\nu_1=1$.

2. For all $k=2$, 3, ...,

$$s_{k-1}=\nu_1+\nu_2+\ldots+\nu_{k-1}\geq \nu_k-1.$$

A corollary states that a Sequence for which $\nu_1=1$ and $\nu_{k+1}\leq 2\nu_k$ is Complete (Honsberger 1985).

See also Complete Sequence

References

Brown, J. L. Jr. ``Notes on Complete Sequences of Integers.'' Amer. Math. Monthly 68, 557-560, 1961.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 123-130, 1985.