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

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

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

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

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

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

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

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

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

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.

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

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Contribution from From _Hieronymus Fischer, _, Apr 04 2019 : (Start):

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Contribution 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)

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