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

1,1

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link?]

Example

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.

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

Prog

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

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.

Sequence in context: A014344 A239613 A100313 * A321851 A185789 A322617

Adjacent sequences: A091076 A091077 A091078 * A091080 A091081 A091082

Keyword

base,nonn

Author

Chuck Seggelin (barkeep(AT)plastereddragon.com), Dec 18 2003

Extensions

More terms from Michel Marcus, Oct 10 2014

Status

approved