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 A058183 Number of digits in concatenation of first n positive integers. 27
 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 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

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Last modified September 28 02:54 EDT 2023. Contains 365714 sequences. (Running on oeis4.)