A121719 - OEIS

%I #23 Sep 14 2020 21:47:04

%S 4,6,8,9,20,22,24,26,28,30,33,36,39,40,42,44,46,48,50,55,60,62,63,64,

%T 66,68,69,70,77,80,82,84,86,88,90,93,96,99,100,110,112,114,116,118,

%U 120,121,130,132,134,136,138,140,143,144

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

%C Comments from _Franklin T. Adams-Watters_:

%C "Think of these as polynomials. E.g. 121 is the polynomial n^2+2n+1. There are three cases:

%C "(1) If the coefficients (digits) all have a common factor, the result will be divisible by that factor.

%C "(2) If the polynomial can be factored, the numbers will be composite. n^2+2n+1 = (n+1)^2, so it is always composite.

%C "(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.

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

%C "If none of the above conditions apply, the polynomial will (probably) have prime values."

%C From _Iain Fox_, Sep 02 2020: (Start)

%C 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).

%C 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.

%C 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).

%C All members of A008592 except 1 and 10 are in this sequence.

%C (End)

%H Iain Fox, <a href="/A121719/b121719.txt">Table of n, a(n) for n = 1..10000</a>

%H Iain Fox, <a href="https://math.stackexchange.com/questions/3810150/what-percentage-of-positive-integers-written-in-base-10-are-composite-regardle/">What percentage of positive integers, written in base 10, are composite regardless of what base they are interpreted in?</a>, Math StackExchange, September 2020.

%H Ed Pegg, Jr., <a href="https://mathworld.wolfram.com/BouniakowskyConjecture.html">Bouniakowsky Conjecture</a>.

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/ExtendedRiemannHypothesis.html">Extended Riemann Hypothesis</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky conjecture</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis">Generalized Riemann Hypothesis</a>.

%e 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.

%o (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

%Y Supersequence: A002808.

%Y Cf. A008592, A267509.

%K nonn,easy,base

%O 1,1

%A _Tanya Khovanova_, Sep 08 2006

%E More terms from _Franklin T. Adams-Watters_, Sep 12 2006