A number for which a shortest chain exists (which is also a Hansen chain) is called a Hansen number.
Hansen Number
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References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111-112, 1994.Referenced on Wolfram|Alpha
Hansen NumberCite this as:
Weisstein, Eric W. "Hansen Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HansenNumber.html