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%I #46 Mar 03 2020 05:52:58
%S 0,1,2,3,4,5,6,7,8,9,9,10,11,12,13,14,15,16,17,18,18,19,20,21,22,23,
%T 24,25,26,27,27,28,29,30,31,32,33,34,35,36,36,37,38,39,40,41,42,43,44,
%U 45,45,46,47,48,49,50,51,52,53,54,54,55,56,57,58,59,60,61,62,63,63,64,65,66,67,68,69,70,71,72,72,73,74,75,76,77,78,79,80,81,81,82,83
%N Number of zerofree nonnegative integers <= n.
%C This sequence represents the counting function of A052382 (for indices > 0). The offset is set to zero for compatibility with A324160.
%C For indices 1 < n < 10761677 we have a(n) > A324160(n), for all indices n > 20327615 we have a(n) < A324160(n), i.e., the number of zerofree numbers <= N is smaller than the number of zero containing numbers <= N for sufficiently large N.
%C There are exactly 20 indices for which a(n) = A324160(n).
%C The greatest number n = n_max such that a(n) >= pi(n) [the number of primes <= n] is in the range 1.0075615552421*10^45 < n_max < 1.0075622026833*10^45 (see A324155). Thus, for all indices n > n_max, we have a(n) < pi(n). For n = n_max the number of primes is pi(n) = 9818959014098676479127822164411318257546629.
%C The least number n = n_min such that a(n) <= pi(n) [the number of primes <= n] is in the range 1.0953002073198*10^44 < n_min < 1.0953009588121*10^44 (see A324154). Thus, for all indices n < n_min, we have a(n) > pi(n). For n = n_min the number of primes is pi(n) = 1090995446010964053236424684934590917505180.
%C Differs from A028904 first at a(100)=90 <> A028904(100)=81. - _R. J. Mathar_, Mar 03 2020
%C Differs from A081600 first at a(101)=90 <> A081600(101)=91. - _R. J. Mathar_, Mar 03 2020
%H Hieronymus Fischer, <a href="/A324161/b324161.txt">Table of n, a(n) for n = 0..10000</a>
%F With m := floor(log_10(n)); k := Max_{j | j=1..m and (floor(n/10^j) mod 10)*j = 0} = digit position of the leftmost '0' in n counted from the right (starting with 0), k = 0 if there is no '0' digit; b(n,k) := floor(n/10^k)*10^k:
%F a(n) = n - 1 - Sum_{j=1..m} floor((b(n,k+1)-1)/10^j)*9^(j-1), if k = 0 (valid for n > 9),
%F a(n) = b(n,k) - 1 - Sum_{j=1..m} floor((b(n,k)-1)/10^j)*9^(j-1), if k > 0 (valid for n > 0),
%F a(n) = b(n,k) - 1 + ceiling(fract(n/10))*(1-ceiling(k/(m+1))) - Sum_{j=1..m} floor((b(n,k)-1)/10^j)*9^(j-1) (all k, valid for n > 0).
%F a(n) + A324160(n) = n + 1.
%F a(A052382(n)) = n.
%F A052382(a(n)) <= n, for n > 0.
%F A052382(a(n)) = n, iff n is a zerofree number.
%F a(10*n + k) >= 9*a(n) + k, k=0..9, equality holds, if n is a zerofree number (i.e., contains no '0'-digit).
%F a(10*A052382(n) + k) = 9*n + k, k=0..9, n > 0.
%F Values for special indices:
%F a(k*(10^n - 1)/9 - j) = k*(9^n - 1)/8 - j, n > 0, k = 1, 2, ... 9, j = 0, 1, 2, ... k.
%F a(k*10^n - j) = k*9^n + (9^n - 1)/8 - j, n >= 0, k = 1, 2, ... 10, j = 1, 2, ... 10.
%F a(k*10^n + j) = k*9^n + (9^n - 1)/8 - 1, n > 0, k = 1, 2, ... 10, j = 0, 1, 2, ... (10^(n+1)-1)/9 - 10^n - 1.
%F With: d := log_10(9) = 0.95424250943932...
%F Upper bound:
%F a(n) <= (9*(n+1)^d - 1)/8 - 1,
%F equality holds for n = 10^k - 1, k >= 0.
%F Lower bound:
%F a(n) >= ((9*n + 10)^d - 1)/8 - 1,
%F equality holds for n = (10^k - 1)/9 - 1, k > 0.
%F Asymptotic behavior:
%F a(n) <= (9/8)*n^d*(1 + O(1/n)) - 9/8.
%F a(n) >= (9^d/8)*n^d*(1 + O(1/n)) - 9/8.
%F a(n) = O(n^d) = O(n^0.954242509...).
%F Lower and upper limits:
%F lim inf a(n)/n^d = 9^d/8 = 1.0173931195971..., for n -> infinity.
%F lim sup a(n)/n^d = 9/8, for n -> infinity.
%F From _Hieronymus Fischer_, Apr 04 2019: (Start)
%F Formulas for general bases b > 2:
%F With m := floor(log_b(n)); k := Max_{j | j=1..m and (floor(n/b^j) mod b)*j = 0} = digit position of the leftmost '0' in n counted from the right (starting with 0), k = 0 if there is no '0' digit; b(n,k):= floor(n/b^k)*b^k:
%F a(n) = n - 1 - Sum_{j=1..m} floor((b(n,k+1)-1)/b^j)*(b-1)^(j-1), if k = 0, valid for n > b-1;
%F a(n) = b(n,k) - 1 - Sum_{j=1..m} floor((b(n,k)-1)/b^j)*(b-1)^(j-1), if k > 0, valid for n > 0;
%F a(n) = b(n,k) - 1 + ceiling(fract(n/b))*(1-ceiling(k/(m+1))) - Sum_{j=1..m} floor((b(n,k)-1)/b^j)*(b-1)^(j-1), (all k, valid for n > 0).
%F Formula for base b = 2: a(n) = floor(log_2(n + 1)).
%F With d := d(b) := log(b - 1)/log(b).
%F Upper bound (b = 10 for this sequence):
%F a(n) <= ((b - 1)*(n + 1)^d - 1)/(b - 2) - 1,
%F equality holds for n = b^k - 1, k >= 0.
%F Lower bound (b = 10 for this sequence):
%F a(n) >= (((b - 1)*n + b)^d - 1)/(b - 2) - 1,
%F equality holds for n = (b^k - 1)/(b - 1) - 1, k > 0.
%F Asymptotic behavior (b = 10 for this sequence):
%F a(n) = O(n^d(b)), for b > 2,
%F a(n) = O(log(n)), for b = 2.
%F Lower and upper limits:
%F lim inf a(n)/n^d = (b - 1)^d/(b - 2), for n -> infinity, for b > 2.
%F lim sup a(n)/n^d = (b - 1)/(b - 2), for n -> infinity, for b > 2.
%F In case of b = 2:
%F lim a(n)/log(n) = 1/log(2), for n -> infinity.
%F (End)
%e a(10) = 9, since there are 9 numbers <= 10 which contain no '0'-digit (1, 2, 3, 4, 5, 6, 7, 8 and 9).
%e a(100) = 90.
%e a(10^3) = 819.
%e a(10^4) = 7380.
%e a(10^5) = 66429.
%e a(10^6) = 597870.
%e a(10^7) = 5380839
%e a(10^8) = 48427560.
%e a(10^9) = 435848049.
%e a(10^10) = 3922632450.
%e a(10^20) = 13677373641439044900.
%e a(10^50) = 579799710823512747416018770986323931789870962250 = 5.79799...*10^47.
%e a(10^100) = 2.9881573748536...68682786424500*10^95.
%e a(10^1000) = 1.96635515818798306...435874245000*10^954.
%p A324161 := proc(n)
%p option remember;
%p if n = 0 then
%p 0;
%p else
%p convert(convert(n,base,10),set) ;
%p if 0 in % then
%p procname(n-1) ;
%p else
%p 1+procname(n-1) ;
%p end if;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 03 2020
%o (PARI) a(n) = sum(k=1, n, vecmin(digits(k)) != 0); \\ _Michel Marcus_, Mar 20 2019
%Y Cf. A011540, A052382, A324160, A324154, A324155.
%K nonn,base
%O 0,3
%A _Hieronymus Fischer_, Feb 15 2019
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