A282473 - OEIS

A282473

Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

%I #41 Nov 27 2019 11:27:27

%S 90,189,288,387,486,585,684,783,882,981,990,1089,1188,1287,1386,1485,

%T 1584,1683,1782,1881,1980,1989,2088,2187,2286,2385,2484,2583,2682,

%U 2781,2880,2979,2988,3087,3186,3285,3384,3483,3582,3681,3780,3879,3978,3987,4086,4185,4284,4383,4482,4581,4680,4779,4878,4977,4986

%N Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

%C Based on empirical observations, it can be noted that the difference between consecutive terms is usually 99. However, this breaks down when the hundreds digit is 9, in which case the difference between consecutive terms is 9. This changes back, however.

%H Charles R Greathouse IV, <a href="/A282473/b282473.txt">Table of n, a(n) for n = 1..10000</a>

%H Brilliant.org, <a href="https://brilliant.org/problems/those-are-some-nifty-numbers/?ref_id=1322236">Those are some nifty numbers</a>

%H Australian Mathematics Olympiad, <a href="https://www.amt.edu.au/wp-content/uploads/2019/07/Australian-Scene-combined-2015-1.pdf">Question 2</a>, 2015, p. 84.

%e If k=90, then k-s(k)=81. If k=100, then k-s(k)=99. This is an increasing function, so k-s(k)=90 is unachievable.

%t okQ[n_] := Catch[ Do[ If[ x- Total@ IntegerDigits@ x == n, Throw@ False], {x, n, n+ 9 IntegerLength[n]}]; True]; Select[9 Range[1000], okQ[#] &] (* _Giovanni Resta_, Feb 27 2017 *)

%o (Python)

%o from math import ceil

%o def a(n): #Outputs all numbers less than n which are in the sequence

%o def s(n):

%o r = 0

%o while n:

%o r, n = r + n% 10, n//10

%o return r

%o mult9=[]

%o if n%9==0:

%o for x in range(1, ceil(n/9)+1):

%o mult9.append(9*x)

%o else:

%o for x in range(1, ceil(n/9)):

%o mult9.append(9*x)

%o for y in range(1, ceil(n/10)+1):

%o mult9.remove(10*y-s(10*y))

%o return mult9

%o (PARI) is(n)=for(k=n,n+9*#Str(n)+9, if(k-sumdigits(k)==n, return(0))); n%9==0 \\ _Charles R Greathouse IV_, Feb 27 2017

%Y Cf. A003052, A176995.

%K nonn,base,easy

%O 1,1

%A _Sharvil Kesarwani_, Feb 16 2017