This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A029581 #40 Feb 15 2024 01:53:08
%S A029581 4,6,8,9,44,46,48,49,64,66,68,69,84,86,88,89,94,96,98,99,444,446,448,
%T A029581 449,464,466,468,469,484,486,488,489,494,496,498,499,644,646,648,649,
%U A029581 664,666,668,669,684,686,688,689,694,696,698,699,844,846,848
%N A029581 Numbers in which all digits are composite.
%C A029581 If n is represented as a zerofree base-4 number (see A084544) according to n=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)=4,6,8,9 for k=1..4. - _Hieronymus Fischer_, May 30 2012
%H A029581 Hieronymus Fischer, <a href="/A029581/b029581.txt">Table of n, a(n) for n = 1..10000</a>
%H A029581 Robert Baillie and Thomas Schmelzer, <a href="https://library.wolfram.com/infocenter/MathSource/7166/">Summing Kempner's Curious (Slowly-Convergent) Series</a>, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
%H A029581 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%F A029581 From _Hieronymus Fischer_, May 30 and Jun 25 2012: (Start)
%F A029581 a(n) = Sum_{j=0..m-1} (2*b(j) mod 8 + 4 + floor(b(j)/4) - floor((b(j)+1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
%F A029581 Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.
%F A029581 Special values:
%F A029581 a(1*(4^n-1)/3) = 4*(10^n-1)/9.
%F A029581 a(2*(4^n-1)/3) = 2*(10^n-1)/3.
%F A029581 a(3*(4^n-1)/3) = 8*(10^n-1)/9.
%F A029581 a(4*(4^n-1)/3) = 10^n-1.
%F A029581 a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.
%F A029581 a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.
%F A029581 a(n) > 2*A084544(n).
%F A029581 Lower and upper limits:
%F A029581 lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.
%F A029581 lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.
%F A029581 where 10^log_4(n) = n^1.66096404744...
%F A029581 G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(4 + 6z(j) + 8*z(j)^2 + 9*z(j)^3)/(1-z(j)^4), where z(j) = x^4^j.
%F A029581 Also: g(x) = (1/(1-x))*(4*h_(4,0)(x) + 2*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+1)-1)/3)*(x^(k*4^j)/(1-x^4^(j+1)). (End)
%F A029581 Sum_{n>=1} 1/a(n) = 1.039691381254753739202528087006945643166147087095114911673083135126969046250... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - _Amiram Eldar_, Feb 15 2024
%e A029581 From _Hieronymus Fischer_, May 30 2012: (Start)
%e A029581 a(1000) = 88649.
%e A029581 a(10^4) = 6468989
%e A029581 a(10^5) = 449466489. (End)
%t A029581 Table[FromDigits/@Tuples[{4, 6, 8, 9}, n], {n, 3}] // Flatten (* _Vincenzo Librandi_, Dec 17 2018 *)
%o A029581 (Magma) [n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // _Vincenzo Librandi_, Dec 17 2018
%Y A029581 Cf. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.
%K A029581 nonn,base
%O A029581 1,1
%A A029581 _N. J. A. Sloane_
%E A029581 Offset corrected by _Arkadiusz Wesolowski_, Oct 03 2011