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Search: id:A121719
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| A121719 |
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Strings of digits which are composite regardless of the base in which they are interpreted. Exclude bases in which numbers are not interpretable. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Comments from Franklin T. Adams-Watters:
"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."
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EXAMPLE
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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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CROSSREFS
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Sequence in context: A123710 A075243 A024370 this_sequence A032350 A078337 A046351
Adjacent sequences: A121716 A121717 A121718 this_sequence A121720 A121721 A121722
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KEYWORD
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more,nonn
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AUTHOR
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Tanya Khovanova (tanyakh(AT)yahoo.com), Sep 08 2006
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EXTENSIONS
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More terms from Franklin T. Adams-Watters, Sep 12 2006
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