A091079 - OEIS

%I #17 Apr 22 2021 21:58:59

%S 16,96,416,496,576,2016,2496,2976,10016,10416,12096,12496,14976,50016,

%T 52416,60096,62496,74976,250016,252016,260416,262416,300096,302096,

%U 310496,312496,360576,374976,1250016,1262016,1300416,1312416,1500096,1512096,1550496

%N 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.

%C 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.

%C n/m must be either 2 or 4. - _Robert Israel_, Apr 22 2021

%H Robert Israel, <a href="/A091079/b091079.txt">Table of n, a(n) for n = 1..10000</a>

%H C. Seggelin, <a href="http://www.plastereddragon.com/maths/asortdiv.htm">Numbers Divisible by Digit Permutations</a>. [Broken link?]

%e 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.

%p F:= proc(d) local eq,m,R;

%p   R:= NULL;

%p   for m in [2,4] do

%p     eq:= m*add(a[i]*5^i,i=0..d)-add(a[d-i]*5^i,i=0..d);

%p     R:= R, F1(eq,[],d);

%p   od;

%p sort([R]);

%p end proc:

%p F1:= proc(eq,A,d) local V,s,e1,i1,i2,vlo,R,v1,v2,Vp,Vm,emax,emin;

%p       V:= indets(eq);

%p       if nops(V) = 0 then

%p          if eq = 0 then subs(A,add(a[d-i]*5^i,i=0..d))

%p          else NULL

%p          fi

%p       elif nops(V) = 1 then

%p          s:= solve(eq,V[1]);

%p          if member(s,[$0..4]) then

%p             subs([op(A),V[1]=s],add(a[d-i]*5^i,i=0..d));

%p          fi

%p       else

%p           Vp,Vm:= selectremove(t -> coeff(eq,t)>0, V);

%p          emax:= subs(map(`=`,Vp,4),map(`=`,Vm,0),eq);

%p          if emax < 0 then return NULL fi;

%p          emin:= subs(map(`=`,Vp,0),map(`=`,Vm,4),eq);

%p          if emin > 0 then return NULL fi;

%p            e1:= eq mod 5;

%p          V:= indets(e1);

%p          if nops(V) = 0 then procname(e1/5,A,d)

%p          elif nops(V) = 1 then

%p            s:= msolve(e1, 5);

%p            procname(subs(s,eq)/5, [op(A),op(s)], d)

%p          else

%p            i1:= op(1,V[1]); i2:= op(1,V[2]);

%p            if i1 = 0 or i2 = 0 then vlo:= 1 else vlo:= 0 fi;

%p            R:= NULL;

%p            for v1 from vlo to 4 do

%p              s:= msolve(eval(e1, a[i1]=v1),5);

%p              R:= R, procname(subs(a[i1]=v1, op(s), eq)/5, [op(A),a[i1]=v1,op(s)],d)

%p            od;

%p            R

%p       fi fi

%p end proc:

%p seq(op(F(d)),d=1..8); # _Robert Israel_, Apr 22 2021

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

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

%Y See also A222816, A214927.

%K base,nonn

%O 1,1

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

%E More terms from _Michel Marcus_, Oct 10 2014