

A121719


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


1



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, 110, 112, 114, 116, 118, 120, 121, 130, 132, 134, 136, 138, 140, 143, 144
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OFFSET

1,1


COMMENTS

Comments from Franklin T. AdamsWatters:
"Think of these as polynomials. E.g. 121 is the polynomial n^2+2n+1. There are three cases:
"(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, 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 (probably) have prime values."


LINKS

Table of n, a(n) for n=1..54.


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.


CROSSREFS

Cf. A267509.
Sequence in context: A123710 A075243 A024370 * A267509 A162738 A161600
Adjacent sequences: A121716 A121717 A121718 * A121720 A121721 A121722


KEYWORD

more,nonn,base


AUTHOR

Tanya Khovanova, Sep 08 2006


EXTENSIONS

More terms from Franklin T. AdamsWatters, Sep 12 2006


STATUS

approved



