A372010 a(n) is the n-digit number k such that R(k)/k is maximal for any n-digit number.

1, 19, 109, 1099, 10099, 100999, 1000999, 10009999, 100009999, 1000099999, 10000099999, 100000999999, 1000000999999, 10000009999999, 100000009999999, 1000000099999999, 10000000099999999, 100000000999999999, 1000000000999999999, 10000000009999999999, 100000000009999999999

Offset

1,2

Links

Table of n, a(n) for n=1..21.

Michael S. Branicky, Proof of form of a(n) for OEIS A372010

Index entries for linear recurrences with constant coefficients, signature (11,0,-110,100).

Formula

a(n) = 1 0^(n-1-h) 9^h, where h = floor(n/2) and ^ represents repeated concatenation (see links for proof).

a(n) = 10^(n-1) + 10 ^ floor(n / 2) - 1.

G.f.: x*(1 + 8*x - 100*x^2 + 10*x^3)/((1 - x)*(1 - 10*x)*(1 - 10*x^2)). - Stefano Spezia, Apr 16 2024

Example

a(2) = 19 as k = 19 is the two digit number k that produces the largest ratio R(k)/k = 91/19 of all two-digit numbers.

Mathematica

Table[10^(n-1) + 10^Floor[n/2] - 1, {n, 25}] (* Paolo Xausa, Apr 23 2024 *)

Prog

(PARI) a(n) = 10^(n-1) + 10 ^ (n \ 2) - 1
(Python) def a(n): return 10**(n-1) + 10**(n//2) - 1

Crossrefs

Cf. A004086.

Sequence in context: A144246 A281170 A158715 * A186105 A080442 A033655

Adjacent sequences: A372007 A372008 A372009 * A372011 A372012 A372013

Keyword

nonn,easy

Author

David A. Corneth and Michael S. Branicky, Apr 15 2024

Status

approved