A282473 Multiples of 9 which cannot be expressed as the difference between a natural number k and its digit sum s(k).

90, 189, 288, 387, 486, 585, 684, 783, 882, 981, 990, 1089, 1188, 1287, 1386, 1485, 1584, 1683, 1782, 1881, 1980, 1989, 2088, 2187, 2286, 2385, 2484, 2583, 2682, 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

Offset

1,1

Comments

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.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Brilliant.org, Those are some nifty numbers

Australian Mathematics Olympiad, Question 2, 2015, p. 84.

Example

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.

Mathematica

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 *)

Prog

(Python)
from math import ceil
def a(n): #Outputs all numbers less than n which are in the sequence
    def s(n):
        r = 0
        while n:
            r, n = r + n% 10, n//10
        return r
    mult9=[]
    if n%9==0:
        for x in range(1, ceil(n/9)+1):
            mult9.append(9*x)
    else:
        for x in range(1, ceil(n/9)):
            mult9.append(9*x)
    for y in range(1, ceil(n/10)+1):
        mult9.remove(10*y-s(10*y))
    return mult9
(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

Crossrefs

Cf. A003052, A176995.

Sequence in context: A119896 A179697 A211442 * A044422 A044803 A235081

Adjacent sequences: A282470 A282471 A282472 * A282474 A282475 A282476

Keyword

nonn,base,easy

Author

Sharvil Kesarwani, Feb 16 2017

Status

approved