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%I #37 Nov 16 2022 14:54:00
%S 0,6,8,9,60,66,68,69,80,86,88,89,90,96,98,99,600,606,608,609,660,666,
%T 668,669,680,686,688,689,690,696,698,699,800,806,808,809,860,866,868,
%U 869,880,886,888,889,890,896,898,899,900,906,908,909,960,966,968,969
%N Numbers in which every digit contains at least one loop (version 1).
%C See A001744 for the other version.
%C If n-1 is represented as a base-4 number (see A007090) according to n-1 = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=0,6,8,9 for k=0..3. - _Hieronymus Fischer_, May 30 2012
%H Hieronymus Fischer, <a href="/A001743/b001743.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%F From _Hieronymus Fischer_, May 30 2012: (Start)
%F a(n) = ((b_m(n)+6) mod 9 + floor((b_m(n)+2)/3) - floor(b_m(n)/3))*10^m + Sum_{j=0..m-1} (b_j(n) mod 4 +5*floor((b_j(n)+3)/4) +floor((b_j(n)+2)/4)- 6*floor(b_j(n)/4)))*10^j, where n>1, b_j(n)) = floor((n-1-4^m)/4^j), m = floor(log_4(n-1)).
%F a(1*4^n+1) = 6*10^n.
%F a(2*4^n+1) = 8*10^n.
%F a(3*4^n+1) = 9*10^n.
%F a(n) = 6*10^log_4(n-1) for n=4^k+1,
%F a(n) < 6*10^log_4(n-1), otherwise.
%F a(n) > 10^log_4(n-1) for n>1.
%F a(n) = 6*A007090(n-1), iff the digits of A007090(n-1) are 0 or 1.
%F G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^4^j *(1-x^4^j)* (6 + 8x^4^j + 9(x^2)^4^j)/(1-x^4^(j+1)).
%F Also: g(x) = (x/(1-x))*(6*h_(4,1)(x) + 2*h_(4,2)(x) + h_(4,3)(x) - 9*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*(x^4^j)^k/(1-(x^4^j)^4). (End)
%e a(1000) = 99896.
%e a(10^4) = 8690099.
%e a(10^5) = 680688699.
%t Union[Flatten[Table[FromDigits/@Tuples[{0,6,8,9},n],{n,3}]]] (* _Harvey P. Dale_, Sep 04 2013 *)
%o (PARI) is(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 4, 5, 7])==0 \\ _Felix Fröhlich_, Sep 09 2019
%Y Cf. A007090, A046034, A029581, A084984, A017042, A001744, A014261, A014263, A202267, A202268.
%K base,nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E Examples added by _Hieronymus Fischer_, May 30 2012
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