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A245491 The least x > 0 such that x < the number of zero digits in the base-n expansions of the numbers 1 through x.

%I A245491 #35 Jun 13 2015 09:31:10
%S A245491 9,87,1068,16022,284704,5834024,135430302,3511116537,100559404366,
%T A245491 3152738985032,107400330425888,3950024143546665,155996847068247395,
%U A245491 6584073072068125453,295764262988176583800,14088968131538370019982,709394716006812244474473
%N A245491 The least x > 0 such that x < the number of zero digits in the base-n expansions of the numbers 1 through x.
%C A245491 If the function zeros(n,b) returns the number of zeros in the numbers 1 through n in base b, then:
%C A245491 zeros(2,2) = zeros_in(10) = 1.
%C A245491 zeros(4,2) = zeros_in(10,100) = 3.
%C A245491 zeros(5,2) = zeros_in(10,100,101) = 4.
%C A245491 zeros(6,2) = zeros_in(10,100,101,110) = 5.
%C A245491 zeros(8,2) = zeros_in(10,100,101,110,1000) = 8.
%C A245491 zeros(9,2) = zeros_in(10,100,101,110,1000,1001) = 10.
%C A245491 Therefore 9 < zeros(9,2) and 9 is the first entry in the list.
%H A245491 Anthony Sand, <a href="/A245491/b245491.txt">Table of n, a(n) for n = 2..100</a>
%e A245491 9 < zero(9,base=2) = 10.
%e A245491 87 < zero(87,3) = 88.
%e A245491 1068 < zero(1068,4) = 1069.
%e A245491 100559404366 < zero(100559404366,10) = 100559404367.
%t A245491 a245491[n_Integer] := Module[{x = 0, z = 0},
%t A245491 While[x >= z, x++; z += Count[IntegerDigits[x, n], 0]]; x]; Map[a245491, Range[2, 12]] (* _Michael De Vlieger_, Aug 06 2014 *)
%o A245491 (PARI) /* formula for calculating n such that zero(n) > n, zero(n-1) <= (n-1) */
%o A245491 {estimate(x,b) = m1=b; est=x\b; nn=est; while(nn>0, d=nn%b; m2 = nn\b; if(d==0, est+=(x%m1)+1; if(m2>0, m2--)); est+=m1*m2; m1*=b; nn=nn\b); return(est)}
%o A245491 {bmin=2; bmx=20; for(bs=bmin,bmx, ni=bs^bs; n=bs+1; ez1=0; ez2=0; until(ez1>n && ez2<=n-1, ez = estimate(n,bs); if(n>=ez, n+=ni, n-=ni; if(ni>1, ni=ni\bs)); ez1 = estimate(n,bs); ez2 = estimate(n-1,bs)); print1(n,", ")) } \\ _Anthony Sand_, Aug 11 2014
%Y A245491 Cf. A164935.
%K A245491 nonn,base
%O A245491 2,1
%A A245491 _Anthony Sand_, Jul 24 2014