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A202267 Numbers in which all digits are noncomposites (1, 2, 3, 5, 7) or 0. 16

%I

%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 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.

%F Contribution 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)

%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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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)