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A058183 Number of digits of concatenation of first n positive integers. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

Eric Weisstein's World of Mathematics, Smarandache Number

FORMULA

a(n) = (n+1)*floor[log10(10n)]-(10^floor[log10(10n)]-1)/(10-1) = a(n-1)+floor[log10(10n)] = A055642(A007908(n)).

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

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

MATHEMATICA

IntegerLength /@ FoldList[#2 + #1 10^IntegerLength[#2] &, Range[50]] (* Eric W. Weisstein, Nov 06 2015 *)

Table[With[{d = Floor[Log10[n]] + 1}, (n + 1) d - (10^d - 1)/9], {n, 50}] (* Eric W. Weisstein, Nov 04 2015 *)

Table[With[{d = IntegerLength[n]}, (n + 1) d - (10^d - 1)/9], {n, 50}] (* Eric W. Weisstein, Nov 06 2015 *)

Length /@ Flatten /@ IntegerDigits /@ Flatten /@ Rest[FoldList[List, {}, Range[20]]] (* Eric W. Weisstein, Nov 04 2015 *)

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

CROSSREFS

Sequence in context: A247945 A225580 A071980 * A080676 A033061 A088380

Adjacent sequences:  A058180 A058181 A058182 * A058184 A058185 A058186

KEYWORD

base,easy,nonn

AUTHOR

Henry Bottomley, Nov 17 2000

STATUS

approved

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Last modified May 29 15:03 EDT 2017. Contains 287247 sequences.