A117804 Natural position of n in the string 12345678910111213....

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

Offset: 1

Author

Warut Roonguthai, Jul 23 2007

Keywords

• nonn
• base

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

Examples

			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)
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A116700.

Cf. A055640, A055641, A055642, A102669-A102685.

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)