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A029581 All digits are composite. 10

%I

%S 4,6,8,9,44,46,48,49,64,66,68,69,84,86,88,89,94,96,98,99,444,446,448,

%T 449,464,466,468,469,484,486,488,489,494,496,498,499,644,646,648,649,

%U 664,666,668,669,684,686,688,689,694,696,698,699,844,846,848

%N All digits are composite.

%C 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 Hieronymus Fischer, <a href="/A029581/b029581.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 and Jun 25 2012: (Start)

%F 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 Also: a(n) = Sum_{j=0..m-1} (A010877(2*b(j)) + 4 + A002265(b(j)) - A002265(b(j)+1))*10^j.

%F Special values:

%F a(1*(4^n-1)/3) = 4*(10^n-1)/9.

%F a(2*(4^n-1)/3) = 2*(10^n-1)/3.

%F a(3*(4^n-1)/3) = 8*(10^n-1)/9.

%F a(4*(4^n-1)/3) = 10^n-1.

%F a(n) < 4*(10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k > 0.

%F a(n) < 4*A084544(n), equality holds iff all digits of A084544(n) are 1.

%F a(n) > 2*A084544(n).

%F Lower and upper limits:

%F lim inf a(n)/10^log_4(n) = 1/10*10^log_4(3) = 0.62127870, for n --> inf.

%F lim sup a(n)/10^log_4(n) = 4/9*10^log_4(3) = 2.756123868970, for n --> inf.

%F where 10^log_4(n) = n^1.66096404744...

%F 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 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)).

%F (End)

%e From _Hieronymus Fischer_, May 30 2012: (Start)

%e a(1000) = 88649.

%e a(10^4) = 6468989

%e a(10^5) = 449466489. (End)

%t Table[FromDigits/@Tuples[{4, 6, 8, 9}, n], {n, 3}] // Flatten (* _Vincenzo Librandi_, Dec 17 2018 *)

%o (MAGMA) [n: n in [1..1000] | Set(Intseq(n)) subset [4, 6, 8, 9]]; // _Vincenzo Librandi_, Dec 17 2018

%Y Cf. A002808, A001744, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_

%E Offset corrected by _Arkadiusz Wesolowski_, Oct 03 2011

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Last modified May 23 19:07 EDT 2019. Contains 323528 sequences. (Running on oeis4.)