A066484 Numbers with at least 2 distinct digits and whose "rotations" (including the number itself) are multiples of these digits; repeated digits allowed but digit 0 not allowed.

1113, 1131, 1311, 2226, 2262, 2622, 3111, 3339, 3393, 3933, 6222, 9333, 11133, 11313, 11331, 13113, 13131, 13311, 22266, 22626, 22662, 26226, 26262, 26622, 31113, 31131, 31311, 33111, 33399, 33939, 33993, 39339, 39393, 39933, 62226, 62262, 62622, 66222, 93339, 93393, 93933, 99333, 111333, 111339, 111393

Offset: 1

Author

Sudipta Das (juitech(AT)vsnl.net), Jan 02 2002

Keywords

• base
• nice
• nonn

Comments

"Rotation" of a (multi-digit) number involves taking the first digit of the number and putting it at the end to form a new number. For example, successive rotations of 1234 yield the numbers 2341, 3412 and 4123 (another rotation gives back the original number).

Subsequence of A034838, A052382 and of A139819. - Reinhard Zumkeller, Nov 29 2012

Examples

			The rotations of 137179 are 371791, 717913, 179137, 791371, 913717, 137179; all these are divisible by 1, 3, 7 and 9.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Ken Duisenberg, Puzzle of the Week (Dec 14, 2001), Dividing Rotated Numbers

Programs

Haskell

-- import Data.List (nub, inits, tails)
a066484 n = a066484_list !! (n-1)
a066484_list = filter h [1..] where
   h x = notElem '0' xs && length (nub xs) > 1 &&
         all d (map read $ zipWith (++)
               (tail $ tails xs) (tail $ inits xs)) where xs = show x
   d u = g u where
         g v = v == 0 || mod u d == 0 && g v' where (v', d) = divMod v 10
-- Reinhard Zumkeller, Nov 29 2012

Mathematica

ddQ[n_]:=Module[{idn=IntegerDigits[n]},DigitCount[n,10,0]==0 && Length[Union[idn]]>1 && And@@Flatten[Divisible[#,Union[idn]]&/@ (FromDigits/@Table[RotateRight[idn,i], {i,Length[idn]}])]]; Select[Range[10,200000],ddQ]  (* Harvey P. Dale, Mar 30 2011 *)

PARI

select( {is_A066484(n,d=Set(digits(n)))= d[1] && #d>1 && (d[1]>1||d=d[^1]) && !for(i=0,logint(n,10),n=[1,10^logint(n,10)]*divrem(n,10);[n%x|x<-d]&&return)}, [1..10^5]) \\ M. F. Hasler, Jan 05 2020

Extensions

Corrected and extended by Harvey P. Dale, Mar 30 2011

Definition reworded by M. F. Hasler, Jan 05 2020