A062010 Let n = Sum_i d_i*10^i (0 <= d_i <= 9) be the decimal expansion of n. Then n is in the sequence if Sum_i d_i*b^i divides n for some base b >= 2 in the range max d_i < b < 10.

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 20, 21, 30, 40, 100, 102, 112, 120, 200, 204, 210, 300, 306, 312, 400, 414, 420, 516, 522, 624, 630, 1000, 1010, 1102, 1120, 1232, 1320, 1344, 1422, 2000, 2223, 2240, 2301, 2310, 3000, 3430, 4000, 10000, 10100, 10101, 10356

Offset

1,2

Comments

In other words, a positive number n is in the sequence if when interpreted as a legal number in a smaller base than 10, the result divides n.

The old definition was "Numbers that, when expressed in a smaller base, become a factor of themselves."

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..301

Example

In base 7, 102 is 51, which divides 102, so 102 is in the sequence.
8 is in the sequence because 8 in base 9 is 8, which does divide 8.
But 9 is not in the sequence because there are no bases b between 9 and 10. Likewise all numbers with a 9 in their decimal expansion are excluded.

Mathematica

dtn[L_, b_] := Fold[b#1+#2&, 0, L]; f[n_] := f[n]=Table[dtn[ IntegerDigits[ n, b], 10], {b, 2, 9}]; g[n_] := MemberQ[Flatten[ Map[ f, Divisors[ n]]], n]; Select[Range[20000], g]
rdnQ[n_]:=AnyTrue[n/FromDigits[IntegerDigits[n], Range[ Max[ IntegerDigits[ n]]+1, 9]], IntegerQ]; Select[Range[11000], rdnQ] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 16 2016 *)

Prog

(Haskell)
import Data.List (unfoldr)
a062010 n = a062010_list !! (n-1)
a062010_list = filter f [1..] where
   f x = any (== 0) $ map (mod x) lower where
       lower = map bas [1 + a054055 x .. 9]
       bas b = foldl (\v d -> b*v + d) 0 bas10
       bas10 = reverse $ unfoldr dig x where
          dig n = if n== 0 then Nothing else Just $ swap $ divMod n 10
-- Reinhard Zumkeller, Oct 09 2011

Crossrefs

Cf. A054055.

Sequence in context: A320319 A263363 A061920 * A071218 A215775 A363149

Adjacent sequences: A062007 A062008 A062009 * A062011 A062012 A062013

Keyword

base,easy,nice,nonn

Author

Erich Friedman, Jun 27 2001

Extensions

Offset corrected by Reinhard Zumkeller, Oct 09 2011

Definition clarified by N. J. A. Sloane, Jul 16 2016

Status

approved