A267939 Number x = concat(MSD(x),b), where MSD = A000030 stands for Most Significant Digit, such that MSD(x)*b is equal to the reverse of x.

351, 621, 886, 920781, 3524751, 338752611, 35247524751, 920780120781, 920879219781, 3387524752611, 3526124738751, 338738752612611, 352475247524751, 33875247524752611, 35247387526124751, 35261247524738751, 920780120780120781, 920780219879120781, 920879120780219781, 920879219879219781

Offset: 1

Author

Paolo P. Lava, Jan 22 2016

Keywords

• base
• nonn

Comments

If we consider numbers x = concat(a,b), where a has two digits, such that a*b is equal to the reverse of x, the first terms are 425322, 44235301, 119910901, ...

Terms of the form 3(5247)*51, i.e. 351, 3524751, 35247524751, ..., form an infinite subsequence. - Robert Israel, Jan 28 2016

Other infinite sequences of terms include 92078(012078)*1 and 33875(2475)*2611. - Robert Israel, Jan 31 2016

Examples

			3*51 = 153;
6*21 = 126;
3*524751 = 1574253.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..415

Crossrefs

Cf. A000030, A004086.

Programs

Maple

T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,b,n; for n from 1 to q do a:=n mod 10; b:=trunc(n/10^ilog10(n));
if (a=1 and b>1) or (a=6 and (b=2 or b=4 or b=6 or b=8)) or (b=5 and (a=3 or a=5 or a=7 or a=9)) then
if T(n)=b*(n mod 10^ilog10(n)) then print(n); fi; fi; od; end: P(10^10);
# alternative:
N:= 20: # to get all terms with at most N digits.
extend:= proc(d,psol,eqs)
  local peqs, cvars, bvars, ncs, res,T, cs, ceqs, sol, svals;
  peqs:= subs(psol, eqs);
  cvars,bvars:= selectremove(t -> op(0,t) = 'c',indets(peqs));
  ncs:= nops(cvars);
  res:= NULL;
  if ncs >= 1 then
    T:= combinat:-cartprod([[$0..d-1]$ncs]);
    while not T[finished] do
      cs:= T[nextvalue]();
      cs:= seq(cvars[i]=cs[i],i=1..ncs);
      ceqs:= subs(cs,peqs);
      sol:= solve(ceqs,bvars); svals:= map(rhs,sol);
      if indets(svals) <> {} then error("Oops: %1",svals) fi;
      if svals::set(nonnegint) and max(svals) <= 9 then
        res:= res, [op(psol), cs, op(sol)];
      fi
    od
  else
    sol:= solve(peqs,bvars);
    svals:= map(rhs,sol);
    if indets(svals) <> {} then error("Oops: %1",svals) fi;
    if svals::set(nonnegint) and max(svals) <= 9 then
        res:= [op(psol), op(sol)];
    fi
  fi;
  [res]
end proc:
G:= proc(d,n)
     local eqs, i, rs, b0s;
     eqs:= [d*b[0] - d - 10*c[0],
            seq(d*b[i]+c[i-1] - b[n-i] - 10*c[i], i=1..n-2),
            d*b[n-1] + c[n-2] - b[1] - 10*b[0]];
     b0s:= [msolve(eqs[1] mod 10,10)];
     rs:= select(t -> (map(rhs,t))::set(nonnegint),
         map(t -> t union solve(eval(eqs[1],t),{c[0]}),b0s));
     for i from 1 to floor(n/2) do
        rs:= map(s -> op(extend(d,s,{eqs[i+1],eqs[-i]})), rs);
     od;
     sort(map(s -> d*10^n + subs(s, add(10^i*b[i],i=0..n-1)), rs));
end proc:
A:= NULL;
for n from 2 to N-1 do
  for d from 3 to 9 do
    res:= G(d,n);
    if res <> [] then
      A:= A, op(res);
    fi
  od
od:
A; # Robert Israel, Feb 01 2016

Mathematica

Select[Range@ 4000000, First[#] FromDigits@ Rest@ # == FromDigits@ Reverse@ # &@ IntegerDigits@ # &] (* Michael De Vlieger, Jan 29 2016 *)

Extensions

a(7) to a(20) from Robert Israel, Feb 01 2016