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) = 2 + Sum_{j = 1..m} floor((b(n,k+1)-1)/10^j)*9^(j-1), if k = 0 (valid for n > 9),
a(n) = 2 + n mod 10^k + Sum_{j = 1..m} floor((b(n,k)-1)/10^j)*9^(j-1), if k > 0 (valid for n > 0),
a(n) = 2 + n mod 10^k - 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).
A011540(a(n)) = n, iff n is a zero-containing number.
a(10*n + k) <= 9*a(n) + n - 8, k = 0..9, equality holds for k = 9, and also, if n is a zerofree number (i.e., contains no '0'-digit).
Values for special indices:
a(k*(10^n-1)/9 - j) = k*(8*10^n - 9*9^n + 1)/72 + 1, n > 0, k = 1..9, j = 0..k.
a(k*10^n - j) = k*(10^n - 9^n) + 1 - (9^n - 1)/8, n >= 0, k = 1..10, j = 1..10.
a(10^n) = 10^n + 2 - (9^(n+1) - 1)/8, n > 0.
a(k*10^n + j) = k*(10^n - 9^n) + j + 2 - (9^n - 1)/8, n > 0, k = 1..9, 0 <= j < (10^(n+1)-1)/9 - 10^n.
With: d := log_10(9) = 0.95424250943932...
Upper bound:
a(n) <= n + 2 - ((9*n + 10)^d - 1)/8,
equality holds for n = (10^k - 1)/9 - 1, k > 0.
Lower bound:
a(n) >= n + 2 - (9*(n + 1)^d - 1)/8,
equality holds for n = 10^k - 1, k >= 0.
Asymptotic behavior:
a(n) <= n + 2 + (1/8) - (9^d/8)*n^d*(1 + O(1/n)).
a(n) >= n + 2 + (1/8) - (9/8)*n^d*(1 + O(1/n))).
a(n) = n*(1 + O(n^(d-1)) = n*(1 + O(1/n^0.045757490...)).
Lower and upper limits:
lim inf (a(n) - n)/n^d = -9/8, for n -> infinity.
lim sup (a(n) - n)/n^d = -9^d/8 = -1.0173931195971..., for n -> infinity.
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) = 2 + 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) = 2 + n mod b^k + Sum_{j=1..m} floor((b(n,k)-1)/b^j)*(b-1)^(j-1), if k > 0, valid for n > 0;
a(n) = 2 + n mod b^k - 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) = (n + 1 - floor(log_2(n + 1))).
With d := d(b) := log(b - 1)/log(b):
Upper bound (b = 10 for this sequence):
a(n) <= n + 2 - (((b - 1)*n + b)^d - 1)/(b - 2),
equality holds for n = (b^k - 1)/(b - 1) - 1, k > 0.
Lower bound (b = 10 for this sequence):
a(n) >= n + 2 - ((b - 1)*(n + 1)^d - 1)/(b - 2),
equality holds for n = b^k - 1, k >= 0.
Asymptotic behavior (b = 10 for this sequence):
a(n) = n*(1 + O(n^(d(b)-1)), for b > 2,
a(n) = n*(1 + O(log(n)/n)), for b = 2.
Lower and upper limits:
lim inf (a(n) - n)/n^d(b) = -(b - 1)/(b - 2), for n -> infinity, for b > 2.
lim sup (a(n) - n)/n^d(b) = -(b - 1)^d/(b - 2) for n -> infinity, for b > 2.
In case of b = 2:
lim (a(n) - n)/log(n) = -1/log(2), for n -> infinity.
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