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%I #39 Feb 15 2024 01:57:16
%S 0,1,2,3,5,7,10,11,12,13,15,17,20,21,22,23,25,27,30,31,32,33,35,37,50,
%T 51,52,53,55,57,70,71,72,73,75,77,100,101,102,103,105,107,110,111,112,
%U 113,115,117,120,121,122,123,125,127,130,131,132,133,135,137,150
%N Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0.
%C If n-1 is represented as a base-6 number (see A007092) 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)=0,1,2,3,5,7 for k=0..5. - _Hieronymus Fischer_, May 30 2012
%H Hieronymus Fischer, <a href="/A202267/b202267.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 <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)+1) mod 10 + floor((b_m(n)+2)/5) + floor((b_m(n)+1)/5) - 2*floor(b_m(n)/5))*10^m + sum_{j=0..m-1} (b_j(n) mod 6 + floor((b_j(n)+1)/6) + floor((b_j(n)+2)/6) - 2*floor(b_j(n)/6)))*10^j, where n>1, b_j(n)) = floor((n-1-6^m)/6^j), m = floor(log_6(n-1)).
%F a(1*6^n+1) = 1*10^n.
%F a(2*6^n+1) = 2*10^n.
%F a(3*6^n+1) = 3*10^n.
%F a(4*6^n+1) = 5*10^n.
%F a(5*6^n+1) = 7*10^n.
%F a(n) = 10^log_6(n-1) for n=6^k+1, k>0,
%F a(n) < 10^log_6(n-1) else.
%F a(n) = A007092(n-1) iff the digits of A007092(n-1) are <= 3, a(n)>A007092(n-1), else.
%F a(n) <= A084984(n), equality holds if the representation of n-1 as a base-6 number only has digits 0 or 1.
%F G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^6^j *(1-x^6^j)* (1 + 2x^6^j + 3(x^2)^6^j + 5(x^3)^6^j + 7(x^4)^6^j)/(1-x^6^(j+1)).
%F Also: g(x) = (x/(1-x))*(h_(6,1)(x) + h_(6,2)(x) + h_(6,3)(x) + 2*h_(6,4)(x) + 2*h_(6,5)(x) - 7*h_(6,6)(x)), where h_(6,k)(x) = sum_{j>=0} 10^j*x^(k*6^j)/(1-x^6^(j+1)). (End)
%F Sum_{n>=2} 1/a(n) = 4.945325883472729555972742252181522711968119529132581193614012706741310832798... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - _Amiram Eldar_, Feb 15 2024
%e a(1000) = 5353.
%e a(10^4) = 115153
%e a(10^5) = 2070753.
%e a(10^6) = 33233353.
%t Union[Flatten[FromDigits/@Tuples[{0,1,2,3,5,7},3]]] (* _Harvey P. Dale_, Mar 11 2015 *)
%Y Supersequence of A001742 and A046034.
%Y Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202268 (numbers in which all digits are nonprimes excluding 0), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).
%Y Cf. A007092, A001743, A001744, A193238.
%K nonn,base,easy
%O 1,3
%A _Jaroslav Krizek_, Dec 25 2011
%E Examples added by _Hieronymus Fischer_, May 30 2012
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