A151752 a(n) is the unique n-digit number with all digits odd that is divisible by 5^n.

5, 75, 375, 9375, 59375, 359375, 3359375, 93359375, 193359375, 3193359375, 73193359375, 773193359375, 3773193359375, 73773193359375, 773773193359375, 5773773193359375, 15773773193359375, 515773773193359375, 7515773773193359375, 97515773773193359375

Offset: 1

Author

David W. Wilson, Jun 16 2009

Keywords

• nonn
• base

Comments

Another way to phrase the proof of uniqueness: after we take the last n-1 digits to be the previous number in the sequence, all odd possibilities for the first digit give different remainders mod 5. By the pigeonhole principle, exactly one of them generates the required number. - Tanya Khovanova, Jun 18 2009

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..300
• 33rd USAMO 2003, Problem 1
• Index to sequences related to Olympiads.

Crossrefs

Cf. A151753, A151754.

Programs

Magma

v:=[5];
for i in [2..20] do
    for s in [1, 3, 5, 7, 9] do
         v[i]:=s*10^(i-1)+v[i-1];
           if v[i] mod 5^i eq 0 then
                 break;
           end if;
    end for;
end for;
v; // Marius A. Burtea, Mar 18 2019

Maple

a:= proc(n) option remember; local k, l;
      if n=1 then 5
    else l:= a(n-1);
         for k from 1 to 9 by 2
           while (parse(cat(k, l)) mod 5^n)<>0 do od;
         parse(cat(k, l))
      fi
    end:
seq(a(n), n=1..30); # Alois P. Heinz, Jun 18 2009

Mathematica

nxt[n_]:=Module[{x=FromDigits/@(Prepend[IntegerDigits[n],# ]&/@{1,3,5,7,9}),l},l=IntegerLength[n]+1;First[Select[x,Mod[ #,5^l]==0&]]]; NestList[nxt, 5, 25] (* Harvey P. Dale, Jul 06 2009 *)

Formula

a(n) = d(n)*10^(n-1) + a(n-1), where d(n), the leading digit of a(n), is one of the odd digits 1, 3, 5, 7, or 9 (forming the complete set of residues modulo 5) and is uniquely defined by the congruence: d(n) == (-a(n-1) / 10^(n-1)) (mod 5). - Max Alekseyev

Extensions

More terms from Max Alekseyev, Jun 17 2009

Further terms from Alois P. Heinz, Jun 18 2009

More terms from Harvey P. Dale, Jul 06 2009