  It is known that in some cases a Random Access Machine (RAM) benefits from
having an additional input that is an arbitrary number, satisfying only the
criterion of being sufficiently large. This is known as the ARAM model. We
introduce a new type of RAM, which we refer to as the Arbitrary Sequence RAM
(ASRAM), that generalises the ARAM by allowing the generation of additional
arbitrary large numbers at will during execution time. We characterise the
power contribution of this ability under several RAM variants.
  In particular, we demonstrate that an arithmetic ASRAM is more powerful than
an arithmetic ARAM, that a sufficiently equipped ASRAM can recognise any
language in the arithmetic hierarchy in constant time (and more, if it is given
more time), and that, on the other hand, in some cases the ASRAM is no more
powerful than its underlying RAM.
