A055641 -id:A055641 - cp's OEIS frontend

A117804 Natural position of n in the string 12345678910111213....

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 1

Warut Roonguthai, Jul 23 2007

The number of digits necessary to write down all the numbers 0, 1, 2, ..., n-1. Thus, the partial sums of A055642 are given by a(n+1). - Hieronymus Fischer, Jun 08 2012
From Daniel Forgues, Mar 21 2013: (Start)
From n = 10^0 + 1 to 10^1: a(n) - a(n-1) = 1 (9 * 10^0 terms);
From n = 10^1 + 1 to 10^2: a(n) - a(n-1) = 2 (9 * 10^1 terms);
From n = 10^2 + 1 to 10^3: a(n) - a(n-1) = 3 (9 * 10^2 terms);
...
From n = 10^k + 1 to 10^(k+1): a(n) - a(n-1) = k+1 (9 * 10^k terms). (End)
By the "number of digits" definition, a(n) = 1 + A058183(n-1) for n > 1. - David Fifield, Jun 02 2019

			12 begins at the 14th place in 12345678910111213... (we are ignoring "early bird" occurrences here, cf. A116700), so a(12) = 14.
From _Daniel Forgues_, Mar 21 2013: (Start)
a(10^1) = 10. (1*10^1 - 0)
a(10^2) = 190. (2*10^2 - 10)
a(10^3) = 2890. (3*10^3 - 110)
a(10^4) = 38890. (4*10^4 - 1110)
a(10^5) = 488890. (5*10^5 - 11110)
a(10^6) = 5888890. (6*10^6 - 111110)
...
a(10^k) = k*10^k - R_k + 1, R_k := k-th repunit (cf. A002275)
(the number of digits necessary to write down the numbers 0..10^k-1). (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A116700.

Cf. A055640, A055641, A055642, A102669-A102685.

a(n) = d*n + 1 - (10^d - 1)/9 where d is the number of decimal digits in n, i.e., d = floor(log_10(n)) + 1.
From Hieronymus Fischer, Jun 08 2012: (Start)
a(n) = Sum_{j=0..n-1} A055642(j).
a(n) = 1 + A055642(n-1)*n - (10^A055642(n-1)-1)/9.
a(n) = 1 + A055642(n)*n - (10^A055642(n)-1)/9.
a(10^n) = (9*n-1)*(10^n-1)/9 + n + 1. (This is the total number of digits necessary to write down all the numbers with <= n places.)
G.f.: g(x) = x/(1-x) + (x/(1-x)^2)*Sum_{j>=0} x^10^j; corrected by Ilya Gutkovskiy, Jan 09 2017 (End)

A027870 Number of zero digits in 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 2, 3, 2, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 1, 1, 0, 1, 3, 3, 1, 0, 1, 1, 1, 0, 0, 2, 4, 2, 0, 2, 3, 1, 1, 0, 3, 5, 3, 3, 4, 2, 3, 4, 1, 1, 4, 5, 5, 6, 6, 7, 5, 5

Offset: 0

N. J. A. Sloane

I conjecture that any value x = 0, 1, 2, ... occurs only a finite number of times N(x) = 36, 41, 31, 34, 25, 32, 37, 23, 43, 47, 33, ... in this sequence, for the last time at well defined indices i(x) = 86, 229, 231, 359, 283, 357, 475, 476, 649, 733, 648, ... - M. F. Hasler, Jul 09 2025

			2^31 = 2147483648 so a(31) = 0 and 2^42 = 4398046511104 so a(42) = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from _Harry J. Smith_, Oct 27 2009)

Cf. A000079 (powers of 2), A007377 (2^n has no zeros).
Similar for other digits: A065712 (1's), A065710 (2's), A065714 (3's), A065715 (4's), A065716 (5's), A065717 (6's), A065718 (7's), A065719 (8's), A065744 (9's).
Cf. A031146 (index of first appearance of n in this sequence), A094776 (index of last occurrence of digit n in powers of 2).
Cf. A305932 (table with n in row a(n)).

a027870 = a055641 . a000079  -- Reinhard Zumkeller, Apr 30 2013

Table[ Count[ IntegerDigits[2^n], 0], {n, 0, 100} ]
DigitCount[2^Range[0,110],10,0] (* Harvey P. Dale, Nov 20 2011 *)

A027870(n)=#select(d->!d,digits(2^n)) \\ M. F. Hasler, Jun 14 2018

def A027870(n):
    return str(2**n).count('0') # Chai Wah Wu, Feb 14 2020

a(n) = A055641(A000079(n)). - Reinhard Zumkeller, Apr 30 2013

a(A007377(n)) = 0; A224782(n) <= a(n). - Reinhard Zumkeller, Apr 30 2013

Edited by M. F. Hasler, Jul 09 2025

A102669 Number of digits >= 2 in decimal representation of n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 1, 1, 1

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007088 (numbers in base 2). - Bernard Schott, Feb 19 2023

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

Cf. A027868, A054899, A055640, A055641, A102670 - A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102670.

Cf. A000120, A000788, A007088, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=2 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..116); # Emeric Deutsch, Feb 23 2005

Table[Total@ Take[DigitCount@ n, {2, 9}], {n, 0, 104}] (* Michael De Vlieger, Aug 17 2017 *)

Contribution from Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 4/5) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(2*10^j) - x^(10*10^j))/(1 - x^10^(j+1)).
General formulas for the number of digits >= d in the decimal representations of n, where 1 <= d <= 9:
a(n) = Sum_{j=1..m+1} (floor(n/10^j + (10-d)/10) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(d*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A102685 Partial sums of A055640.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123

Offset: 0

N. J. A. Sloane, Feb 03 2005

nonn

The total number of nonzero digits occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

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

Cf. A027868, A054899, A055640, A055641, A102669-A102684, A117804, A122840, A122841, A160093, A160094, A196563, A196564.

Cf. A000120, A000788, A023416, A059015 (for base 2).

From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor((n/10^j)+0.9)*(2n + 2 + (0.8 - floor((n/10^j)+0.9))*10^j) - floor(n/10^j)*(2n + 2 - (floor(n/10^j)+1) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A055640(n) + (1/2)*Sum_{j=1..m+1} ((8*floor((n/10^j)+0.9)/10 + floor(n/10^j))*10^j - (floor((n/10^j)+0.9)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 9*m*10^(m-1). (This is the total number of nonzero digits occurring in all the numbers with <= m digits.)
G.f.: g(x) = (1/(1-x)^2) * Sum_{j>=0} (x^10^j - x^(10*10^j))/(1-x^10^(j+1)). (End)

A160094 a(n) = 1 + A122840(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Anonymous, May 01 2009

a(n) is the Levenshtein distance from the decimal expansion of n - 1 to the decimal expansion of n. For example, to convert "9" to "10", substitute "0" for "9" and insert "1". Since two such operations are required, a(10) = 2. See the analogous A091090 (binary expansion) and A115777 (full definition). - Rick L. Shepherd, Mar 25 2015

			a(160) = 2 because the last nonzero digit of 160 (counting from left to right), when 160 is written in base 10, is 6, and that 6 occurs 2 digits from the right in 160.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A054899, A055640, A055641, A102669, A122840, A122841, A160093, A196563, A196564, A091090, A115777.

IntegerExponent[Range[150]]+1 (* Harvey P. Dale, Feb 06 2015 *)

From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_10(n)), frac(x) = x-floor(x):
a(n) = Sum_{j=0..m} (1 - ceiling(frac(n/10^j))).
a(n) = m + 1 + Sum_{j=1..m} (floor(-frac(n/10^j))).
a(n) = 1 + A054899(n) - A054899(n-1).
G.f.: g(x) = (x/(1-x)) + Sum_{j>0} x^10^j/(1-x^10^j). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 10/9. - Amiram Eldar, Jul 10 2023
a(n) = A122840(10*n). - R. J. Mathar, Jun 28 2025

Name simplified by Jon E. Schoenfield, Feb 26 2014

A160093 Number of digits in n, excluding any trailing zeros.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3

Offset: 1

Anonymous, May 01 2009

			a(1060000) = 3 because discarding the trailing zeros from 1060000 leaves 106, which is a 3-digit number.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A004151, A054899, A055640, A055641, A055642, A102669, A122840, A122841, A160094, A196563, A196564.

lnzd[n_]:=Module[{spl=Last[Split[IntegerDigits[n]]]},If[!MemberQ[ spl,0], IntegerLength[n], IntegerLength[n]-Length[spl]]]; Array[lnzd,110] (* Harvey P. Dale, Jun 05 2013 *)
Table[IntegerLength[n] - IntegerExponent[n, 10], {n, 100}] (* Amiram Eldar, Sep 14 2020 *)

a(n)=if(n==0,1,#digits(n/10^valuation(n,10))) \\ Joerg Arndt, Jan 11 2017

a(n)=logint(n,10)+1-valuation(n,10) \\ Charles R Greathouse IV, Jan 12 2017

def A160093(n):
     return len(str(int(str(n)[::-1]))) # Indranil Ghosh, Jan 11 2017

From Hieronymus Fischer, Jun 08 2012: (Start)
With m = floor(log_10(n)), frac(x) = x-floor(x):
a(n) = 1 + Sum_{j=0..m} ceiling(frac(n/10^j)).
a(n) = 1 - Sum_{j=1..m} (floor(-frac(n/10^j))).
a(n)= A055642(n) + A054899(n-1) - A054899(n).
G.f.: (x/(1-x)) + (1/(1-x))*Sum_{j>0} x^(10^j+1)*(1 - x^(10^j-1))/(1-x^10^j). (End)
a(n) = A055642(A004086(n)). - Indranil Ghosh, Jan 11 2017
a(n) = A055642(A004151(n)). - Amiram Eldar, Sep 14 2020

Simpler definition and changed example from Jon E. Schoenfield, Feb 15 2014

A168046 Characteristic function of zerofree numbers in decimal representation.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Dec 01 2009

a(A052382(n)) = 1; a(A011540(n)) = 0;
a(n) = A000007(A055641(n));
not the same as A168184: a(n)=A168184(n) for n<=100.
a(A007602(n)) = a(A038186(n)) = 1. - Reinhard Zumkeller, Apr 07 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Zerofree
Index entries for characteristic functions

Cf. A052382, A217096, A011540.

a168046 = fromEnum . ch0 where
   ch0 x = x > 0 && (x < 10 || d > 0 && ch0 x') where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015, Apr 07 2011

Map[Boole[FreeQ[IntegerDigits[#], 0]] &, Range[0, 100]] (* Paolo Xausa, May 06 2024 *)

a(n) = A057427(A010879(n)) * (if n<10 then 1 else a(A059995(n))).
From Hieronymus Fischer, Jan 23 2013: (Start)
a(n) = A057427(A007954(n)) = sign(dp_10(n)).
where dp_10(n) digital product of n in base 10.
a(n) = 1 - A217096(n).
a(n) = 1 - sign(A055641(n)).
g(x) = x(1-x^9)/((1-x)(1-x^10))(1 + sum_{j>=1} (x^((10^j-10)/9) - x^10^j)/(1-x^10^(j+1)))).
g(x) = 1/(1-x) - g_A217096(x), where g_A217096(x) is the g.f. of A217096.
(End)

A102683 Number of digits 9 in decimal representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Feb 03 2005

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007095, A011539, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564, A235049.
Cf. A102684 (partial sums).
Cf. A000120, A000788, A023416, A059015 (for base 2).

a102683 =  length . filter (== '9') . show
-- Reinhard Zumkeller, Dec 29 2011

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=9 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..116); # Emeric Deutsch, Feb 23 2005

a[n_] := DigitCount[n, 10, 9]; Array[a, 100, 0] (* Amiram Eldar, Jul 24 2023 *)

a(A007095(n)) = 0; a(A011539(n)) > 0. - Reinhard Zumkeller, Dec 29 2011
From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 1/10) - floor(n/10^j)), where m=floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(9*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)
a(A235049(n)) = 0. - Reinhard Zumkeller, Apr 16 2014

More terms from Emeric Deutsch, Feb 23 2005

A119291 Total number of zero digits in first 10^n primes.

Original entry on oeis.org

0, 9, 191, 3303, 46188, 557005, 6481183, 76292782, 881025347, 9763247930, 106864564286, 1162019145892

Offset: 1

Enoch Haga, May 13 2006

Count the zero digits in the first 10^n primes.

			a(2)=9 since there are 9 zero digits in the first 100 primes.

Cf. A119290, A119292-A119300.

A055641 := proc(n) local a,d ; a := 0 ; for d in convert(n,base,10) do if d = 0 then a := a+1 ; fi ; od: a ; end: p := 2: n := 1: c :=0 : nsw := 10 : while true do n := n+1 ; p := nextprime(p) ; c := c+A055641(p) ; if n = nsw then print(c) ; nsw := 10*nsw ; fi ; od: # R. J. Mathar, May 30 2008

Table[Count[IntegerDigits[Prime[Range[10^n]]], 0, 2], {n, 6}] (* Robert Price, May 02 2019 *)

my(x=10, i=0, j=0); forprime(p=1, , j++; my(d=digits(p)); i+=#setintersect(vecsort(d), vector(#d, t, 0)); if(j==x, print1(i, ", "); x=10*x)) \\ Felix Fröhlich, May 02 2019

a(n) = Sum_{j=1..10^n} A055641(A000040(j)). - R. J. Mathar, May 30 2008

a(8)-a(11) from Robert Price, Nov 05 2013

a(12) from Marek Hubal, Mar 04 2019

A292730 Numbers in which 0 outnumbers all other digits together.

Original entry on oeis.org

0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 10001, 10002, 10003, 10004, 10005, 10006, 10007, 10008, 10009, 10010, 10020, 10030, 10040, 10050, 10060, 10070, 10080, 10090, 10100, 10200, 10300, 10400, 10500, 10600, 10700, 10800, 10900, 11000

Offset: 1

Halfdan Skjerning, Sep 22 2017

Subset of A292450.
Numbers n such that A055641(n) > (A055642(n)/2). - Felix Fröhlich, Sep 22 2017
Also numbers whose median of the digits is equal to 0. - Stefano Spezia, Oct 04 2023

			100 has more 0's than any other digit, whereas both 1001 and 1002 have as many other digits as 0's.

Cf. A292449, A292450, A292451, A292452, A292453, A292454, A292455, A292456, A292457, A292458, A292731, A292732, A292733, A292734, A292735, A292736, A292737, A292738, A292739.

Cf. A055641, A055642.

Select[Range[0, 11000], Total@ #1 < First@ #2 & @@ TakeDrop[DigitCount@ #, 9] &] (* Michael De Vlieger, Sep 22 2017 *)

a055641(n)=if(n, n=digits(n); sum(i=2, #n, n[i]==0), 1) \\ after Charles R Greathouse IV
is(n) = a055641(n) > (#Str(n)/2) \\ Felix Fröhlich, Sep 22 2017

cp's OEIS Frontend

A117804 Natural position of n in the string 12345678910111213....

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A027870 Number of zero digits in 2^n.

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A102669 Number of digits >= 2 in decimal representation of n.

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A102685 Partial sums of A055640.

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A160094 a(n) = 1 + A122840(n).

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A160093 Number of digits in n, excluding any trailing zeros.

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A168046 Characteristic function of zerofree numbers in decimal representation.

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