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 A117804 Natural position of n in the string 12345678910111213.... 33
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS Hieronymus Fischer, Table of n, a(n) for n = 1..10000 FORMULA 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) EXAMPLE 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) CROSSREFS Cf. A116700. Cf. A055640, A055641, A055642, A102669-A102685. Sequence in context: A067082 A102685 A032960 * A088235 A064223 A098952 Adjacent sequences:  A117801 A117802 A117803 * A117805 A117806 A117807 KEYWORD nonn,base AUTHOR Warut Roonguthai, Jul 23 2007 STATUS approved

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Last modified November 14 12:33 EST 2019. Contains 329114 sequences. (Running on oeis4.)