A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

16, 96, 416, 496, 576, 2016, 2496, 2976, 10016, 10416, 12096, 12496, 14976, 50016, 52416, 60096, 62496, 74976, 250016, 252016, 260416, 262416, 300096, 302096, 310496, 312496, 360576, 374976, 1250016, 1262016, 1300416, 1312416, 1500096, 1512096, 1550496

Offset: 1

Author

Chuck Seggelin, Dec 18 2003

Keywords

• base
• nonn

Comments

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

n/m must be either 2 or 4. - Robert Israel, Apr 22 2021

Examples

			a(1) = 16 because: 16 in base 5 is 31; 31 reversed is 13; 13 converted back to base 10 is 8 and 16 mod 8 = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link?]

Crossrefs

Cf. A091077 (same in base 3), A091078 (base 4), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

See also A222816, A214927.

Programs

Maple

F:= proc(d) local eq,m,R;
  R:= NULL;
  for m in [2,4] do
    eq:= m*add(a[i]*5^i,i=0..d)-add(a[d-i]*5^i,i=0..d);
    R:= R, F1(eq,[],d);
  od;
sort([R]);
end proc:
F1:= proc(eq,A,d) local V,s,e1,i1,i2,vlo,R,v1,v2,Vp,Vm,emax,emin;
      V:= indets(eq);
      if nops(V) = 0 then
         if eq = 0 then subs(A,add(a[d-i]*5^i,i=0..d))
         else NULL
         fi
      elif nops(V) = 1 then
         s:= solve(eq,V[1]);
         if member(s,[$0..4]) then
            subs([op(A),V[1]=s],add(a[d-i]*5^i,i=0..d));
         fi
      else
          Vp,Vm:= selectremove(t -> coeff(eq,t)>0, V);
         emax:= subs(map(`=`,Vp,4),map(`=`,Vm,0),eq);
         if emax < 0 then return NULL fi;
         emin:= subs(map(`=`,Vp,0),map(`=`,Vm,4),eq);
         if emin > 0 then return NULL fi;
           e1:= eq mod 5;
         V:= indets(e1);
         if nops(V) = 0 then procname(e1/5,A,d)
         elif nops(V) = 1 then
           s:= msolve(e1, 5);
           procname(subs(s,eq)/5, [op(A),op(s)], d)
         else
           i1:= op(1,V[1]); i2:= op(1,V[2]);
           if i1 = 0 or i2 = 0 then vlo:= 1 else vlo:= 0 fi;
           R:= NULL;
           for v1 from vlo to 4 do
             s:= msolve(eval(e1, a[i1]=v1),5);
             R:= R, procname(subs(a[i1]=v1, op(s), eq)/5, [op(A),a[i1]=v1,op(s)],d)
           od;
           R
      fi fi
end proc:
seq(op(F(d)),d=1..8); # Robert Israel, Apr 22 2021

PARI

/* See A091077 and use PARI script with b=5 */

Extensions

More terms from Michel Marcus, Oct 10 2014