Special numbers consecutive A number is said special if it has two or more digits and is a multiple of the sum of its
digits.
(a) Find three consecutive special numbers.
(b) Find four consecutive special numbers.
Well, as I understand it, a special number is for example: $12$ is special because it's a multiple of $ 1 + 2 = 3$
 A: http://oeis.org/A060159
$110,111,112$
$510,511,512,513$
$131052,131053,131054,131055,131056$
are all "special numbers" (more commonly called Harshad or Niven numbers in literature)
C. N. Cooper and R. E. Kennedy proved in a paper submitted to Fibonacci Quarterly in 1993 that there are not sequences of $21$ or more consecutive such numbers.
It is also known that there does exist sequences of $20$ consecutive of these numbers, the smallest known sequence of $20$ consecutive of these numbers being on the order of $10^{44363342786}$.  It is not yet known what the smallest possible such sequence of $20$ consecutive numbers is, but by the well ordering principle it must exist. [$\dagger$]
Even more generally, when considering these numbers represented instead in base $b$, H. G. Grundman in 1994 extended this result to show that there cannot exist sequences of $2b+1$ consecutive numbers for any base $b$, which was later expanded on by Brad Wilson to show that there will however exist sequences of $2b$ consecutive numbers for any base $b$, which was later expanded on even further by T. Cai in 1996 to show that for $b=2$ or $b=3$ there are infinitely many such sequences.
