A324161 Number of zerofree nonnegative integers <= n.

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, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 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

Offset: 0

Author

Hieronymus Fischer, Feb 15 2019

Keywords

• nonn
• base

Comments

This sequence represents the counting function of A052382 (for indices > 0). The offset is set to zero for compatibility with A324160.

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.

There are exactly 20 indices for which a(n) = A324160(n).

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.

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.

Differs from A028904 first at a(100)=90 <> A028904(100)=81. - R. J. Mathar, Mar 03 2020

Differs from A081600 first at a(101)=90 <> A081600(101)=91. - R. J. Mathar, Mar 03 2020

Examples

			a(10) = 9, since there are 9 numbers <= 10 which contain no '0'-digit (1, 2, 3, 4, 5, 6, 7, 8 and 9).
a(100) = 90.
a(10^3) = 819.
a(10^4) = 7380.
a(10^5) = 66429.
a(10^6) = 597870.
a(10^7) = 5380839
a(10^8) = 48427560.
a(10^9) = 435848049.
a(10^10) = 3922632450.
a(10^20) = 13677373641439044900.
a(10^50) = 579799710823512747416018770986323931789870962250 = 5.79799...*10^47.
a(10^100) = 2.9881573748536...68682786424500*10^95.
a(10^1000) = 1.96635515818798306...435874245000*10^954.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A011540, A052382, A324160, A324154, A324155.

Programs

Maple

A324161 := proc(n)
    option remember;
    if n = 0 then
        0;
    else
        convert(convert(n,base,10),set) ;
        if 0 in % then
            procname(n-1) ;
        else
            1+procname(n-1) ;
        end if;
    end if;
end proc: # R. J. Mathar, Mar 03 2020

PARI

a(n) = sum(k=1, n, vecmin(digits(k)) != 0); \\ Michel Marcus, Mar 20 2019

Formula

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:

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),

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),

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).

a(n) + A324160(n) = n + 1.

a(A052382(n)) = n.

A052382(a(n)) <= n, for n > 0.

A052382(a(n)) = n, iff n is a zerofree number.

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).

a(10*A052382(n) + k) = 9*n + k, k=0..9, n > 0.

Values for special indices:

a(k*(10^n - 1)/9 - j) = k*(9^n - 1)/8 - j, n > 0, k = 1, 2, ... 9, j = 0, 1, 2, ... k.

a(k*10^n - j) = k*9^n + (9^n - 1)/8 - j, n >= 0, k = 1, 2, ... 10, j = 1, 2, ... 10.

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.

With: d := log_10(9) = 0.95424250943932...

Upper bound:

a(n) <= (9*(n+1)^d - 1)/8 - 1,

equality holds for n = 10^k - 1, k >= 0.

Lower bound:

a(n) >= ((9*n + 10)^d - 1)/8 - 1,

equality holds for n = (10^k - 1)/9 - 1, k > 0.

Asymptotic behavior:

a(n) <= (9/8)*n^d*(1 + O(1/n)) - 9/8.

a(n) >= (9^d/8)*n^d*(1 + O(1/n)) - 9/8.

a(n) = O(n^d) = O(n^0.954242509...).

Lower and upper limits:

lim inf a(n)/n^d = 9^d/8 = 1.0173931195971..., for n -> infinity.

lim sup a(n)/n^d = 9/8, for n -> infinity.

From Hieronymus Fischer, Apr 04 2019: (Start)

Formulas for general bases b > 2:

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:

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;

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;

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).

Formula for base b = 2: a(n) = floor(log_2(n + 1)).

With d := d(b) := log(b - 1)/log(b).

Upper bound (b = 10 for this sequence):

a(n) <= ((b - 1)*(n + 1)^d - 1)/(b - 2) - 1,

equality holds for n = b^k - 1, k >= 0.

Lower bound (b = 10 for this sequence):

a(n) >= (((b - 1)*n + b)^d - 1)/(b - 2) - 1,

equality holds for n = (b^k - 1)/(b - 1) - 1, k > 0.

Asymptotic behavior (b = 10 for this sequence):

a(n) = O(n^d(b)), for b > 2,

a(n) = O(log(n)), for b = 2.

Lower and upper limits:

lim inf a(n)/n^d = (b - 1)^d/(b - 2), for n -> infinity, for b > 2.

lim sup a(n)/n^d = (b - 1)/(b - 2), for n -> infinity, for b > 2.

In case of b = 2:

lim a(n)/log(n) = 1/log(2), for n -> infinity.

(End)