A081600 -id:A081600 - 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)

A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27, 28, 29

Offset: 0

N. J. A. Sloane, Apr 22 2003

Eswaran observes that n is divisible by 7 iff repeated application of a ends at the number 7.

a(n) is divisible by 7 iff n is divisible by 7: e.g., a(7) = a(14) = a(21) = 7, a(28) = a(35) = a(42) = 14 etc. - Zak Seidov, Mar 19 2014

R. Eswaran, Test of divisibility of the number 7, Abstracts Amer. Math. Soc., 23 (No. 2, 2002), #974-00-5, p. 275.

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

Different from A028898 for n>=100 (e.g. a(111) = 34, A029989(111) = 13).

Cf. A081503, A081594, A081595, A081596, A081597, A081598, A081599, A081600.

A081502 := proc(n)
    local x,y ;
    y := modp(n,10) ;
    x := iquo(n,10) ;
    3*x+y ;
end proc:
seq(A081502(n),n=0..120) ; # R. J. Mathar, Oct 03 2014

Table[n - 7 * Floor[n / 10], {n, 0, 100}] (* Joshua Oliver, Dec 04 2019 *)

a(n) = 3*(n\10) + (n % 10); \\ Michel Marcus, Mar 19 2014

a(n) = [3,1]*divrem(n,10); \\ Kevin Ryde, Dec 04 2019

G.f.: -x*(6*x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x-1) / (x^11-x^10-x+1). - Colin Barker, Mar 19 2014

a(n) = n-7*floor(n/10). - Wesley Ivan Hurt, May 12 2016

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.

cp's OEIS Frontend

A324161 Number of zerofree nonnegative integers <= n.

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A081502 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 3x+y.

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

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