# Demo*graph*ics

A $$k$$-regular graph satisfies the neighborhood diversity condition if it is possible to label each vertex with one of $$1, 2, 3, \ldots, k$$ such that for each vertex, all its neighbors have different labels.

Determine the sum of all $$n$$ satisfying $$1 \le n \le 1000$$ such that the $$n$$-hypercube graph satisfies the neighborhood diversity condition.


Clarification:

• In a $$k$$-regular graph, each vertex has $$k$$ neighbors.
• An $$n$$-hypercube graph is the graph of an $$n$$-dimensional hypercube. In other words, its vertices are elements of $$\{0,1\}^n$$ (that is, $$n$$-tuples with each component 0 or 1), and two vertices are adjacent if and only if they differ in exactly one component.
• The labeling is not necessarily a proper vertex coloring: two adjacent vertices may have the same label.
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