  Although 10^230 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) <= n^2 - 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 log^2 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*10^12 digits. We have only a probabilistic argument that such a k
exists for all n.
