all-base composites

[franktaw at netscape.net]

First, think of these as polynomials.  E.g., 121  is the polynomial 
n^2+2n+1.
There are three cases you have to look at:
(1) If the coefficients (digits) all have a common factor, the result 
will be
     divisible by that factor.
(2) If the polynomial can be factored, the numbers will be composite.
     n^2+2n+1 = (n+1)^2, so it is always composite.
(3) Otherwise, you need to look at the polynomial modulo primes up
     to its degree.  For example, 112 (n^2+n+2, degree 2) modulo 2 is
     always 0, so it is always divisible by 2.

Note that condition (1) is really a special case of condition (2), 
where one
of the factors is a constant.

If none of the above conditions apply, the polynomial will have prime
values.  I don't think this has actually been proved, but it is 
certainly true
up to any size you will encounter.  You can add a search for a prime
value if you want.  Search up to base 1000 or so, and generate some
kind of error message if you don't find a prime.

The continuation after 100 is 110, 112, 114, 116, 118, 120, 121, 130,
132, 134, 136, 138, 140, 143, 144, ....

By the way, this sequence should have the "base" keyword.

Franklin T. Adams-Watters


-----Original Message-----
From: tanyakh at TanyaKhovanova.com

Hello seqfans,

I recently submitted a sequence I really like. I am also suggesting a 
name for
it: all-base composites.

I am not sure how to code this sequence. Can you help?


A121719  Strings of digits which are composite regardless of the base 
in which
they are interpreted. Exclude bases in which numbers are not 
interpretable.

  4, 6, 8, 9, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 42, 44, 46, 48, 
50, 55, 60,
62, 63, 64, 66, 68, 69, 70, 77, 80, 82, 84, 86, 88, 90, 93, 96, 99, 100

  EXAMPLE  String 55 in every base in which it is interpretable is 
divisible by
5. String 1001 in base a is divisible by a+1. Hence 55 and 1001 both 
belong to
this sequence.

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