A028904 -id:A028904 - cp's OEIS frontend

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

Hieronymus Fischer, Feb 15 2019

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

			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.

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

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

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

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

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)

A081600 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 9x+y.

Original entry on oeis.org

Offset: 0

N. J. A. Sloane, Apr 22 2003

More than the usual number of terms are displayed in order to distinguish this from some closely related sequences. - N. J. A. Sloane, Mar 22 2014

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Different from A028904.

k:=9; [n-(10-k)*Floor(n/10): n in [0..150]]; // Bruno Berselli, Jun 24 2014

f1:=proc(n) local x,y;
y:= (n mod 10);
x:=(n-y)/10;
9*x+y;
end;
[seq(f1(n),n=0..200)];

my(n, x, y); vector(500, n, y=(n-1)%10; x=(n-1-y)\10; 9*x+y) \\ Colin Barker, Jun 23 2014

a(n)=n - n\10 \\ Charles R Greathouse IV, Sep 01 2015

G.f.: x*(x^2 +x +1)*(x^6 +x^3 +1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 24 2014

a(n) = n - floor(n/10). - Bruno Berselli, Jun 24 2014

A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 0

Reinhard Zumkeller, Apr 23 2003

A010879(n)=T(n,0);
A076314(0)=T(0,0), A076314(n)=T(n,1) for n>0;
A028897(n)=T(n,n) for n<=1, A028897(n)=T(n,2) for n>1;
A028898(n)=T(n,n) for n<=2, A028898(n)=T(n,3) for n>2;
A028899(n)=T(n,n) for n<=3, A028899(n)=T(n,4) for n>3;
A028900(n)=T(n,n) for n<=4, A028900(n)=T(n,5) for n>4;
A028901(n)=T(n,n) for n<=5, A028901(n)=T(n,6) for n>5;
A028902(n)=T(n,n) for n<=6, A028902(n)=T(n,7) for n>6;
A028903(n)=T(n,n) for n<=7, A028903(n)=T(n,8) for n>7;
A028904(n)=T(n,n) for n<=8, A028904(n)=T(n,9) for n>8;
T(n,n) = n for n<=9, T(n,10) = n for n>9;
A083292(n) = T(n,n).

			From _Paolo Xausa_, May 22 2024: (Start)
Triangle begins:
   [0] 0;
   [1] 1, 1;
   [2] 2, 2, 2;
   [3] 3, 3, 3, 3;
   [4] 4, 4, 4, 4, 4;
   [5] 5, 5, 5, 5, 5, 5;
   [6] 6, 6, 6, 6, 6, 6, 6;
   [7] 7, 7, 7, 7, 7, 7, 7, 7;
   [8] 8, 8, 8, 8, 8, 8, 8, 8, 8;
   [9] 9, 9, 9, 9, 9, 9, 9, 9, 9, 9;
  [10] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
  ... (End)

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).

Table[k*Floor[n/10] + Mod[n, 10], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, May 22 2024 *)

Offset changed to 0 by Paolo Xausa, May 22 2024

A239092 Prefix overlap of dictionary consisting of decimal expansions of 0 through n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110

Offset: 1

N. J. A. Sloane, Mar 22 2014

The prefix overlap between two words is the length of their longest common prefix.
The prefix overlap of a dictionary is the sum of the prefix overlaps between successive words.
Partial sums of A076489.
More than the usual number of terms are displayed in order to distinguish this from some closely related sequences.

Rodica Simion and Herbert S. Wilf, The distribution of prefix overlap in consecutive dictionary entries, SIAM J. Algebraic Discrete Methods, 7(1986), no. 3, 470--475. MR0844051.

Cf. A238845, A239091, A076489.

Different from A081600 and A028904.

A332690 Sum of all numbers in bijective base-9 numeration with digit sum n.

Original entry on oeis.org

0, 1, 12, 124, 1248, 12496, 124992, 1249984, 12499968, 124999936, 1249999862, 12499999623, 124999998144, 1249999984364, 12499999840480, 124999998308464, 1249999981991936, 12499999808733888, 124999997974967808, 1249999978624935680, 12499999774999871588

Offset: 0

Alois P. Heinz, Feb 19 2020

Different from A016134.

			a(2) = 12 = 2 + 10 = 2_bij9 + 11_bij9.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bijective numeration
Index entries for linear recurrences with constant coefficients, signature (10,1,-8,-17,-26,-35,-44,-53,-62,-81,-72,-63,-54,-45,-36,-27,-18,-9).

Cf. A007953, A016134, A028904, A052382, A211072, A214676, A332691.

b:= proc(n) option remember; `if`(n=0, [1, 0], add((p->
      [p[1], p[2]*9+p[1]*d])(b(n-d)), d=1..min(n, 9)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..23);

G.f.: (Sum_{j=1..9} j*x^j) / ((B(x) - 1) * (9*B(x) - 1)) with B(x) = Sum_{j=1..9} x^j.
a(n) = A028904(A332691(n)).
a(n) = A016134(n-1) for n = 1..9.

cp's OEIS Frontend

A324161 Number of zerofree nonnegative integers <= n.

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A081600 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 9x+y.

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A083291 Triangular array read by rows: T(n,k) = k*floor(n/10) + n mod 10, 0<=k<=n.

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A239092 Prefix overlap of dictionary consisting of decimal expansions of 0 through n.

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A332690 Sum of all numbers in bijective base-9 numeration with digit sum n.

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