A200980 - OEIS

%I #29 Jun 04 2016 06:59:57

%S 1,2,4,33,5,46,6,2,2,3,2,2,6

%N Concatenate the digits of the natural numbers from 1 to n in order to build up two numbers x and y that minimize the ratio x/y > 0, an integer (leading zeros not admitted).

%C For n=8 and n=9 we have 12 possible different fractions:

%C n=8 -> 3456/1728, 3528/1764, 3564/1782, 3654/1827, 4356/2178, 4716/2358, 5436/2718, 5634/2817, 7128/3564, 7164/3582, 8352/4176, 8712/4356.

%C n=9 -> 13458/6729, 13584/6792, 13854/6927, 14538/7269, 14586/7293, 14658/7329, 15384/7692, 15846/7923, 15864/7932, 18534/9267, 18546/9273, 18654/9327. - _Arie Groeneveld_, Nov 25 2011

%C Examples for n=10..13:  a(10) = 3 = 161427/53809, a(11) = 2 = 1141826/570913, a(12) = 2 = 11418226/5709113, and a(13) = 6 = 114312678/19052113. - _Giovanni Resta_, May 31 2016

%e Starting with a(1)=1 we have a(2)=2/1=2, a(3)=12/3=4, a(4)=132/4=33, a(5)=215/43=5, a(6)= 2346/51= 46, a(7)= 3426/571=6, a(8)= 3456/1728 = 2, a(9)= 13458/6729=2.

%p with(combinat,permute);

%p P:=proc(i)

%p local a,c,d,j,k,m,ok,n,t,v,x,y;

%p v:=[1,2]; t:=2; lprint(1,1); lprint(2,2);

%p for n from 3 to i do

%p   c:=n;

%p   for j from 1 to floor(1+evalf(log10(n))) do

%p     t:=t+1; v:=[op(v),c-10*trunc(c/10)]; c:=trunc(c/10);

%p   od;

%p   if (t mod 2)=1 then a:=(t+1)/2; else a:=t/2; fi;

%p   c:=permute(v); d:=nops(c); c:=op(c); m:=10^13; ok:=0;

%p   while ok=0 do

%p     for j from 1 to d do

%p       x:=0; for k from 1 to a do x:=10*x+c[j][k]; od;

%p       y:=0; for k from a+1 to t do y:=10*y+c[j][k]; od;

%p       if x>y then if trunc(x/y)=x/y then ok:=1; if x/y<m then m:=x/y; if m=2 then break; fi; fi; fi; fi;

%p     od;

%p     if ok=0 then a:=a+1; fi;

%p   od;

%p   lprint(n,m);

%p od;

%p end:

%p P(10);

%K nonn,base,hard,more

%O 1,2

%A _Paolo P. Lava_, Nov 25 2011

%E a(5) corrected by _Arie Groeneveld_, Nov 25 2011

%E a(6)-a(9) from _Claudio Meller_, Nov 25 2011

%E a(10)-a(13) from _Giovanni Resta_, May 31 2016