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A168046 Characteristic function of zerofree numbers in decimal representation.

%I A168046 #28 Feb 16 2025 08:33:11
%S A168046 0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,
%T A168046 1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,
%U A168046 1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0
%N A168046 Characteristic function of zerofree numbers in decimal representation.
%C A168046 a(A052382(n)) = 1; a(A011540(n)) = 0;
%C A168046 a(n) = A000007(A055641(n));
%C A168046 not the same as A168184: a(n)=A168184(n) for n<=100.
%C A168046 a(A007602(n)) = a(A038186(n)) = 1. - _Reinhard Zumkeller_, Apr 07 2011
%H A168046 Reinhard Zumkeller, <a href="/A168046/b168046.txt">Table of n, a(n) for n = 0..10000</a>
%H A168046 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zerofree.html">Zerofree</a>
%H A168046 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A168046 a(n) = A057427(A010879(n)) * (if n<10 then 1 else a(A059995(n))).
%F A168046 From _Hieronymus Fischer_, Jan 23 2013: (Start)
%F A168046 a(n) = A057427(A007954(n)) = sign(dp_10(n)).
%F A168046 where dp_10(n) digital product of n in base 10.
%F A168046 a(n) = 1 - A217096(n).
%F A168046 a(n) = 1 - sign(A055641(n)).
%F A168046 g(x) = x(1-x^9)/((1-x)(1-x^10))(1 + sum_{j>=1} (x^((10^j-10)/9) - x^10^j)/(1-x^10^(j+1)))).
%F A168046 g(x) = 1/(1-x) - g_A217096(x), where g_A217096(x) is the g.f. of A217096.
%F A168046 (End)
%t A168046 Map[Boole[FreeQ[IntegerDigits[#], 0]] &, Range[0, 100]] (* _Paolo Xausa_, May 06 2024 *)
%o A168046 (Haskell)
%o A168046 a168046 = fromEnum . ch0 where
%o A168046    ch0 x = x > 0 && (x < 10 || d > 0 && ch0 x') where (x', d) = divMod x 10
%o A168046 -- _Reinhard Zumkeller_, May 10 2015, Apr 07 2011
%Y A168046 Cf. A052382, A217096, A011540.
%K A168046 base,nonn
%O A168046 0,1
%A A168046 _Reinhard Zumkeller_, Dec 01 2009