[seqfan] Re: A090709 Decimal primes whose decimal representation in base 6 is also prime. -- rebasing

[Marc LeBrun]

>="Veikko Pohjola" <veikko at nordem.fi>
> Should there be some unification of the terminology?

Yes, and here's a possible starting point:

Once upon a time we discussed canonizing the idea of "rebasing" which covers
many of these situations.

The definition of "rebasing n from a to b" is roughly "expand n in powers of
a, then replace a with b".

Thus A090709 might be described succinctly as "primes that when rebased from
10 to 6 are prime".

A major benefit of adopting and using this concept is that it clarifies that
ALL sequences in the OEIS are ALWAYS sequences of INTEGER values that are
ALWAY written in DECIMAL.

In particular it establishes that the values are never in any other base,
nor are they digit strings or other flavors of non-numerical objects.

After all, it's the "On-line Encyclopedia of INTEGER Sequences".  And it's
customary for integers to be "written in decimal" (that redundancy, repeated
in 400+ entries, flags that *some* clarification might be in order!<;-).

Rebasing has many other pleasant ramifications in addition to eliminating
lots of awkward, inconsistent, imprecise and ambiguous circumlocutions.

For instance in defining a transform it relieves concerns about what base to
use to interpret the input values.

Admittedly it takes a little getting used to the idea that, for example, the
concise definition of A007088 is "numbers rebased from 2 to 10", but I think
the advantages in increased precision are well worth it.

A side-benefit is that rebasing is an operation with widespread interesting
and useful applications.  For example rebasing from 10 to 1 sums the digits,
and rebasing from 10 to -10 reverses them (mod a left-shift).

More generally it encourages a standardized viewpoint for expanding integers
into polynomials and evaluating polynomials back into integers at a base.

For example rebasing A014580 from 2 to 10 gives A058943.  Rebasing to X
gives the actual polynomials, which never appear directly in the OEIS, but
are perfectly reasonable in definitions or computations with numerical I/O.

And so on...

Anyway, back in the day I was writing, say 10[n]6, to mean "rebase n from 10
to 6" from a psychedelic analogy with notations like GF2[n](X).  Franklin T.
Adams-Watters proposed n_10->6 which is more suggestive.  Ultimately the
notation discussion trailed off inconclusively; there are now entries with a
diversity of notations, conventions and terminology.

Yes, this should all be unified.

--MLB