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Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Florian Luca, Augustine O. Munagi (2014)

Colloquium Mathematicae

We note that every positive integer N has a representation as a sum of distinct members of the sequence d ( n ! ) n ≥ 1 , where d(m) is the number of divisors of m. When N is a member of a binary recurrence u = u ₙ n ≥ 1 satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

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