Three Cousins of Recaman's Sequence
Abstract
Although 10230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) A(n), n >= 3 is defined as follows. Start with n, and add n+1, n+2, n+3, ..., stopping after adding n+k if the sum n + (n+1) + ... + (n+k) is divisible by n+k+1. Then A(n)=k. We determine A(n) and show that A(n) <= n2 - 2n - 1. (ii) B(n), n >= 1 is a multiplicative analog of A(n). Start with n, and successively multiply by n+1, n+2, ..., stopping after multiplying by n+k if the product n(n+1)...(n+k) is divisible by n+k+1. Then B(n)=k. We conjecture that log2 B(n) = (1/2 + o(1)) log n loglog n. (iii) The third sequence, C(n), n >= 1, is the most interesting, because the most mysterious. Concatenate the decimal digits of n, n+1, n+2, ... until the concatenation n||n+1||...||n+k is divisible by n+k+1. Then C(n)=k. If no such k exists we set C(n)=-1. We have found k for all n <= 1000 except for two cases. Some of the numbers involved are quite large. For example, C(92) = 218128159460, and the concatenation 92||93||...||(92+C(92)) is a number with about 2*1012 digits. We have only a probabilistic argument that such a k exists for all n.