Complete Sequence

A sequence of numbers  is complete if every positive integer  is the sum of some subsequence of , i.e., there exist  or 1 such that

(Honsberger 1985, pp. 123-126). The Fibonacci numbers are complete. In fact, dropping one number still leaves a complete sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). The sequence of primes with the element  prepended,

is complete, even if any number of primes each  are dropped, as long as the dropped terms do not include two consecutive primes (Honsberger 1985, pp. 127-128). This is a consequence of Bertrand's postulate.

REFERENCES:

Brown, J. L. Jr. "Unique Representations of Integers as Sums of Distinct Lucas Numbers." Fib. Quart. 7, 243-252, 1969.

Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer for Fibonacci Numbers. XII." Fib. Quart. 11, 317-331, 1973.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.