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A023705 Numbers with no 0's in base-4 expansion.

%I A023705 #53 Jun 29 2025 11:08:09
%S A023705 1,2,3,5,6,7,9,10,11,13,14,15,21,22,23,25,26,27,29,30,31,37,38,39,41,
%T A023705 42,43,45,46,47,53,54,55,57,58,59,61,62,63,85,86,87,89,90,91,93,94,95,
%U A023705 101,102,103,105,106,107,109,110,111,117,118,119,121,122,123
%N A023705 Numbers with no 0's in base-4 expansion.
%C A023705 A032925 is the intersection of this sequence and A023717; cf. A179888. - _Reinhard Zumkeller_, Jul 31 2010
%H A023705 Reinhard Zumkeller, <a href="/A023705/b023705.txt">Table of n, a(n) for n = 1..10000</a>
%H A023705 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%F A023705 G.f. g(x) satisfies g(x) = (x+2*x^2+3*x^3)/(1-x^3) + 4*(x+x^2+x^3)*g(x^3). - _Robert Israel_, Oct 04 2018
%p A023705 R:= [1,2,3]: A:= 1,2,3:
%p A023705 for i from 1 to 4 do
%p A023705   R:= map(t -> (4*t+1,4*t+2,4*t+3), R);
%p A023705   A:= A, op(R);
%p A023705 od:
%p A023705 A; # _Robert Israel_, Oct 04 2018
%t A023705 Select[ Range[ 120 ], (Count[ IntegerDigits[ #, 4 ], 0 ]==0)& ]
%t A023705 Select[Range[200],DigitCount[#,4,0]==0&] (* _Harvey P. Dale_, Dec 23 2015 *)
%o A023705 (Haskell)
%o A023705 a023705 n = a023705_list !! (n-1)
%o A023705 a023705_list = iterate f 1 where
%o A023705    f x = 1 + if r < 3 then x else 4 * f x'
%o A023705          where (x', r) = divMod x 4
%o A023705 -- _Reinhard Zumkeller_, Mar 06 2015, Oct 19 2011
%o A023705 (PARI) isok(n) = vecmin(digits(n, 4)); \\ _Michel Marcus_, Jul 04 2015
%o A023705 (Magma) [n: n in [1..130] | not 0 in Intseq(n,4)]; // _Vincenzo Librandi_, Oct 04 2018
%o A023705 (C)
%o A023705 #include <stdint.h>
%o A023705 uint32_t a_next(uint32_t a_n) { return (a_n + 1) | ((a_n & (a_n + 0xaaaaaaab)) >> 1); } /* _Falk Hüffner_, Jan 22 2022 */
%o A023705 (Python)
%o A023705 from sympy import integer_log
%o A023705 def A023705(n):
%o A023705     m = integer_log(k:=(n<<1)+1,3)[0]
%o A023705     return sum(1+(k-3**m)//(3**j<<1)%3<<(j<<1) for j in range(m)) # _Chai Wah Wu_, Jun 27 2025
%Y A023705 Cf. A003754, A007090.
%Y A023705 Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A248910 (base 6), A255805 (base 8), A255808 (base 9), A052382 (base 10).
%Y A023705 Cf. A100968 (subsequence).
%K A023705 nonn,base,easy
%O A023705 1,2
%A A023705 _Olivier Gérard_