A181736 - OEIS

%I #9 Mar 30 2012 18:53:06

%S 1,2,2,3,1,0,1,0,0,0,0,0,0,0,0

%N The number of integers in base 2n such that all digits are used exactly once (so length is 2n) and for each m<=2n the base 2n integer consisting of the first m digits is divisible by m.

%C The unique base 10 number is 3816547290: so 3 is divisible by 1, 38 by 2, 381 by 3, 3816 by 4 and so on. Of course the last digit must be 0. It isn't too hard to show that there are none when the base is odd, and not too hard to show that there are none when the base is of the form 2m(2m-1), for m>1. A computer search found the unique number in base 14 and showed that there were no more up to base 28. 30=6*5 is, of course, of the form 2m(2m-1). I do not know whether there are any more.

%C According to the comment to A111456, no other such numbers up to base 40.

%e a(1)=1 because the only number base 2 satisfying the condition is 10. a(2)=2 because the two in base 4 are 1230 and 3210.

%Y The numbers are listed in A111456.

%K nonn,base,more

%O 1,2

%A _David Gauld_