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A058183
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Number of digits in concatenation of first n positive integers.
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19
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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
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OFFSET
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1,2
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COMMENTS
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Or, total number of digits in numbers from 1 through n.
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LINKS
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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
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FORMULA
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a(n) = (n+1)*floor(log_10(10*n)) - (10^floor(log_10(10*n))-1)/(10-1) = a(n-1) + floor(log_10(10*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
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EXAMPLE
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a(12) = 15 since 123456789101112 has 15 digits.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A007908, A053064, A055642.
Sequence in context: A331009 A225580 A071980 * A322341 A080676 A033061
Adjacent sequences: A058180 A058181 A058182 * A058184 A058185 A058186
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KEYWORD
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base,easy,nonn
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AUTHOR
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Henry Bottomley, Nov 17 2000
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STATUS
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approved
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