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%I A014263 #76 Aug 04 2024 06:40:58
%S A014263 0,2,4,6,8,20,22,24,26,28,40,42,44,46,48,60,62,64,66,68,80,82,84,86,
%T A014263 88,200,202,204,206,208,220,222,224,226,228,240,242,244,246,248,260,
%U A014263 262,264,266,268,280,282,284,286,288,400,402,404,406,408,420,422,424
%N A014263 Numbers that contain even digits only.
%C A014263 The set of real numbers between 0 and 1 that contain no odd digits in their decimal expansion has Hausdorff dimension log 5 / log 10.
%C A014263 Integers written in base 5 and then doubled (in base 10). - _Franklin T. Adams-Watters_, Mar 15 2006
%C A014263 The carryless mod 10 "even" numbers (cf. A004529) sorted and duplicates removed. - _N. J. A. Sloane_, Aug 03 2010.
%C A014263 Complement of A007957; A196564(a(n)) = 0; A103181(a(n)) = 0. - _Reinhard Zumkeller_, Oct 04 2011
%C A014263 If n-1 is represented as a base-5 number (see A007091) 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)=0,2,4,6,8 for k=0..4. - _Hieronymus Fischer_, Jun 03 2012
%D A014263 K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 19.
%H A014263 Reinhard Zumkeller, <a href="/A014263/b014263.txt">Table of n, a(n) for n = 1..10000</a>
%H A014263 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%H A014263 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%F A014263 A045888(a(n)) = 0. - _Reinhard Zumkeller_, Aug 25 2009
%F A014263 a(n) = A179082(n) for n <= 25. - _Reinhard Zumkeller_, Jun 28 2010
%F A014263 From _Hieronymus Fischer_, Jun 06 2012: (Start)
%F A014263 a(n) = ((2*b_m(n)) mod 8 + 2)*10^m + Sum_{j=0..m-1} ((2*b_j(n)) mod 10)*10^j, where n>1, b_j(n) = floor((n-1-5^m)/5^j), m = floor(log_5(n-1)).
%F A014263 a(1*5^n+1) = 2*10^n.
%F A014263 a(2*5^n+1) = 4*10^n.
%F A014263 a(3*5^n+1) = 6*10^n.
%F A014263 a(4*5^n+1) = 8*10^n.
%F A014263 a(n) = 2*10^log_5(n-1) for n=5^k+1,
%F A014263 a(n) < 2*10^log_5(n-1), else.
%F A014263 a(n) > (8/9)*10^log_5(n-1) n>1.
%F A014263 a(n) = 2*A007091(n-1), iff the digits of A007091(n-1) are 0 or 1.
%F A014263 G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^5^j *(1-x^5^j)* (2+4x^5^j+ 6(x^2)^5^j+ 8(x^3)^5^j)/(1-x^5^(j+1)).
%F A014263 Also: g(x) = 2*(x/(1-x))*Sum_{j>=0} 10^j*x^5^j * (1-4x^(3*5^j)+3x^(4*5^j))/((1-x^5^j)(1-x^5^(j+1))).
%F A014263 Also: g(x) = 2*(x/(1-x))*(h_(5,1)(x) + h_(5,2)(x) + h_(5,3)(x) + h_(5,4)(x) - 4*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)
%F A014263 a(5*n+i-4) = 10*a(n) + 2*i for n >= 1, i=0..4. - _Robert Israel_, Apr 07 2016
%F A014263 Sum_{n>=2} 1/a(n) = A194182. - _Bernard Schott_, Jan 13 2022
%e A014263 a(1000) = 24888.
%e A014263 a(10^4) = 60888.
%e A014263 a(10^5) = 22288888.
%e A014263 a(10^6) = 446888888.
%p A014263 a:= proc(m) local L,i;
%p A014263   L:= convert(m-1,base,5);
%p A014263   2*add(L[i]*10^(i-1),i=1..nops(L))
%p A014263 end proc:
%p A014263 seq(a(i),i=1..100); # _Robert Israel_, Apr 07 2016
%t A014263 Select[Range[450], And@@EvenQ[IntegerDigits[#]]&] (* _Harvey P. Dale_, Jan 30 2011 *)
%o A014263 (Haskell)
%o A014263 a014263 n = a014263_list !! (n-1)
%o A014263 a014263_list = filter (all (`elem` "02468") . show) [0,2..]
%o A014263 -- _Reinhard Zumkeller_, Jul 05 2011
%o A014263 (Magma) [n: n in [0..424] | Set(Intseq(n)) subset [0..8 by 2]];  // _Bruno Berselli_, Jul 19 2011
%o A014263 (Python)
%o A014263 from sympy.ntheory.digits import digits
%o A014263 def a(n): return int(''.join(str(2*d) for d in digits(n, 5)[1:]))
%o A014263 print([a(n) for n in range(58)]) # _Michael S. Branicky_, Jan 13 2022
%o A014263 (Python)
%o A014263 from itertools import count, islice, product
%o A014263 def agen(): # generator of terms
%o A014263     yield 0
%o A014263     for d in count(1):
%o A014263         for first in "2468":
%o A014263             for rest in product("02468", repeat=d-1):
%o A014263                 yield int(first + "".join(rest))
%o A014263 print(list(islice(agen(), 58))) # _Michael S. Branicky_, Jan 13 2022
%o A014263 (PARI) a(n) = 2*fromdigits(digits(n-1, 5), 10); \\ _Michel Marcus_, Nov 04 2022
%Y A014263 Subsequence of A059708.
%Y A014263 Cf. A061810, A061811, A007091, A014261, A046034, A052382, A084544, A089581, A084984, A017042, A001743, A202267, A202268, A194182, A196563.
%K A014263 nonn,base,easy
%O A014263 1,2
%A A014263 _N. J. A. Sloane_
%E A014263 Examples and crossrefs added by _Hieronymus Fischer_, Jun 06 2012

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