A058183 Number of digits in concatenation of first n positive integers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset

1,2

Comments

Or, total number of digits in numbers from 1 through n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

GeeksforGeeks, Count total number of digits from 1 to n

Eric Weisstein's World of Mathematics, Smarandache Number

Formula

a(n) = (n+1)*floor(log_10(10*n)) - (10^floor(log_10(10*n))-1)/(10-1).

a(n) = a(n-1) + floor(log_10(10*n)).

a(n) = A055642(A007908(n)).

a(n) = A055642(A053064(n)). - Reinhard Zumkeller, Oct 10 2008

a(n) ~ n log_10 n + O(n). In particular lim inf (n log_10 n - a(n))/n = (1+log(10/9)+log(log(10)))/log(10) and the corresponding lim sup is 10/9. - Charles R Greathouse IV, Sep 19 2012

G.f.: (1-x)^(-2)*Sum_{j>=0} x^(10^j). - Robert Israel, Nov 04 2015

a(n) = b(n)*(n + 1) - (10^b(n) - 19)/9 - 2, where b(n) = A055642(n). - Lorenzo Sauras Altuzarra, May 09 2020

a(n) = A055642(A000422(n)). - Michel Marcus, Sep 11 2021

Example

a(12) = 15 since 123456789101112 has 15 digits.

Maple

a:= proc(n) a(n):= `if`(n=0, 0, a(n-1) +length(n)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2013
a := proc(n) local d; d:=floor(log10(n))+1; (n+1)*d - (10^d-1)/9; end; # N. J. A. Sloane, Feb 20 2020

Mathematica

Length/@ Flatten/@ IntegerDigits/@ Flatten/@ Rest[FoldList[List, {}, Range[70]]] (* Eric W. Weisstein, Nov 04 2015 *)
Table[With[{d = IntegerLength[n]}, (n+1) d - (10^d -1)/9], {n, 70}] (* Eric W. Weisstein, Nov 06 2015 *)
IntegerLength/@ FoldList[#2 + #1 10^IntegerLength[#2] &, Range[70]] (* Eric W. Weisstein, Nov 06 2015 *)
Accumulate[ IntegerLength@ # & /@ Range @ 70] (* Robert G. Wilson v, Jul 31 2018 *)

Prog

(PARI) a(n)=my(t=log(10*n+.5)\log(10)); n*t+t-10^t\9 \\ Charles R Greathouse IV, Sep 19 2012
(PARI) a(n) = sum(k=1, n, #digits(k)); \\ Michel Marcus, Jan 01 2017
(Python)
def A058183(n): return (n+1)*(s:=len(str(n))) - (10**s-1)//9 # Chai Wah Wu, May 02 2023

Crossrefs

Cf. A000422, A007908, A053064, A055642.

Sequence in context: A331009 A225580 A071980 * A322341 A080676 A033061

Adjacent sequences: A058180 A058181 A058182 * A058184 A058185 A058186

Keyword

base,easy,nonn

Author

Henry Bottomley, Nov 17 2000

Status

approved