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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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2
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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
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OFFSET
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1,1
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COMMENTS
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"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."
lim_{k->infinity} (1/k)*Sum_{i=1..k} a_c(i) > .3 if it exists, where a_c(n) is the characteristic function of a(n) (1 if n is in a(n), otherwise 0).
If the Bunyakovsky conjecture is true, the list of reasons a number is in this sequence detailed by Franklin T. Adams-Watters above is a complete list.
If the Bunyakovsky conjecture and the Extended Riemann Hypothesis are true, the above limit equals 4340435807/13235512500 = 0.3279386... (proof by Ravi Fernando in link by Iain Fox).
All members of A008592 except 1 and 10 are in this sequence.
(End)
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LINKS
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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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PROG
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(PARI) is(n)=if(n<10, return(!isprime(n)&&n>1)); if(content(n=digits(n))>1, return(1)); if(vecsum(factor(n*=vectorv(#n, i, x^(#n-i)))[, 2])>1, return(1)); forprime(p=2, #n-1, for(x=1, p, if(eval(n)%p, next(2))); return(1)); for(x=vecmax(Vec(n))+1, +oo, if(isprime(eval(n)), return(0))) \\ Iain Fox, Aug 31 2020
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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