A051885 Smallest number whose sum of digits is n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 199, 299, 399, 499, 599, 699, 799, 899, 999, 1999, 2999, 3999, 4999, 5999, 6999, 7999, 8999, 9999, 19999, 29999, 39999, 49999, 59999, 69999, 79999, 89999, 99999, 199999, 299999, 399999, 499999

Offset: 0

Author

Felice Russo, Dec 15 1999

Keywords

• nonn
• easy
• base
• nice

Comments

This is also the list of lunar triangular numbers: A087052 with duplicates removed. - N. J. A. Sloane, Jan 25 2011

Numbers n such that A061486(n) = n. - Amarnath Murthy, May 06 2001

The product of digits incremented by 1 is the same as the number incremented by 1. If a(n) = abcd...(a,b,c,d, etc. are digits of a(n)) {a(n) + 1} = (a+1)*(b+1)(c+1)*(d+1)*..., e.g., 299 + 1 = (2+1)*(9+1)*(9+1) = 300. - Amarnath Murthy, Jul 29 2003

A138471(a(n)) = 0. - Reinhard Zumkeller, Mar 19 2008

a(n+1) = A108971(A179988(n)). - Reinhard Zumkeller, Aug 09 2010, Jul 10 2011

Positions of records in A003132: A080151(n) = A003132(a(n)). - Reinhard Zumkeller, Jul 10 2011

a(n) = A242614(n,1). - Reinhard Zumkeller, Jul 16 2014

A254524(a(n)) = 1. - Reinhard Zumkeller, Oct 09 2015

The slowest strictly increasing sequence of nonnegative integers such that, for any two terms, calculating the difference of their decimal representations requires no borrowing. - Rick L. Shepherd, Aug 11 2017

Links

• Iain Fox, Table of n, a(n) for n = 0..9000 (first 101 terms from Reinhard Zumkeller)
• D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
• A. Murthy, Exploring some new ideas on Smarandache type sets, functions and sequences, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000.
• Index entries for sequences related to dismal (or lunar) arithmetic
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,10,-10).

Crossrefs

Cf. A061104, A061105, A061486, A007953, A067043, A087052.

Numbers of form i*b^j-1 (i=1..b-1, j >= 0) for bases b = 2 through 9: A000225, A062318, A180516, A181287, A181288, A181303, A165804, A140576. - N. J. A. Sloane, Jan 25 2011

Cf. A002283.

Cf. A254524.

Programs

Haskell

a051885 n = (m + 1) * 10^n' - 1 where (n',m) = divMod n 9
-- Reinhard Zumkeller, Jul 10 2011

Magma

[i*10^j-1: i in [1..9], j in [0..5]];

Maple

b:=10; t1:=[]; for j from 0 to 15 do for i from 1 to b-1 do t1:=[op(t1), i*b^j-1]; od: od: t1; # N. J. A. Sloane, Jan 25 2011

Mathematica

a[n_] := (Mod[n, 9] + 1)*10^Floor[n/9] - 1; Table[a[n], {n, 0, 49}](* Jean-François Alcover, Dec 01 2011, after Henry Bottomley *)

PARI

A051885(n) = (n%9+1)*10^(n\9)-1  \\ M. F. Hasler, Jun 17 2012

PARI

first(n) = Vec(x*(x^2 + x + 1)*(x^6 + x^3 + 1)/((x - 1)*(10*x^9 - 1)) + O(x^n), -n) \\ Iain Fox, Dec 30 2017

Python

def A051885(n): return ((n % 9)+1)*10**(n//9)-1 # Chai Wah Wu, Apr 04 2021

Formula

These are the numbers i*10^j-1 (i=1..9, j >= 0). - N. J. A. Sloane, Jan 25 2011

a(n) = ((n mod 9) + 1)*10^floor(n/9) - 1 = a(n-1) + 10^floor((n-1)/9). - Henry Bottomley, Apr 24 2001

a(n) = A037124(n+1) - 1. - Reinhard Zumkeller, Jan 03 2008, Jul 10 2011

G.f.: x*(x^2+x+1)*(x^6+x^3+1) / ((x-1)*(10*x^9-1)). - Colin Barker, Feb 01 2013

Extensions

More terms from James Sellers, Dec 16 1999

Offset fixed by Reinhard Zumkeller, Jul 10 2011