A061797 Smallest k such that k*n has even digits and is a palindrome or becomes a palindrome when 0's are added on the left.

1, 2, 1, 2, 1, 4, 1, 98, 1, 74, 2, 2, 5, 154, 49, 4, 5, 38, 37, 34, 1, 286, 1, 36, 25, 8, 77, 329144, 31, 16, 2, 28, 25, 2, 19, 196, 23, 6, 17, 154, 1, 542, 143, 1602, 1, 148, 18, 6, 88, 14, 4, 824, 77, 8, 164572, 4, 143, 1198, 8, 1154, 1, 1126, 14, 962, 66, 308, 1, 998

Offset: 0

Author

Amarnath Murthy, Jun 17 2001

Keywords

• nonn
• base
• easy
• nice

Comments

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

a(81), if it exists, is greater than 5 million. - Harvey P. Dale, Dec 19 2021

A palindrome is divisible by 81 iff its sum of digits is divisible by 81. Thus a(81) = 688888888628888888886 / 81 = 8504801097146776406, as 688888888868888888886 is the least palindrome with even digits and sum of digits 162. - Robert Israel, Apr 17 2025

Examples

			a(12) = 5 since 5*12 = 60 (i.e., "060") is a palindrome.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..7000 (n = 0 .. 80 from Reinhard Zumkeller)
• Patrick De Geest, Smallest multipliers to make a number palindromic.

Crossrefs

Cf. A050782, A062293 A061674. Values of k*n are given in A062293.

Cf. A014263, A136522, A004151.

Programs

ARIBAS

stop := 500000; for n := 0 to 75 do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := not all_even(m) or m <> int_reverse(m) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;

Haskell

a061797 0 = 1
a061797 n = head [k | k <- [1..], let x = k * n,
                 all (`elem` "02468") $ show x, a136522 (a004151 x) == 1]
-- Reinhard Zumkeller, Feb 01 2012

Maple

epali:= proc(x,d) local L,i;
  L:= convert(x,base,5);
  if d::even then 2*add(L[-i]*(10^(i-1)+10^(d-i)),i=1..d/2)
  else 2*(L[-(d+1)/2]*10^((d-1)/2) + add(L[-i]*(10^(i-1)+10^(d-i)),i=1..(d-1)/2))
  fi
end proc;
Agenda:= {$0..80}:
count:= 0:
for d from 1 while count < 81 do
  E[d]:= [seq(epali(i,d),i=5^(ceil(d/2)-1) .. 5^ceil(d/2)-1)];
  P:= sort([op(E[d]),seq(op(E[k] *~ 10^(d-k)), k=1..d-1)]);
  for x in P do
    Q:= select(t -> x mod t = 0, Agenda);
    if Q <> {} then
      count:= count + nops(Q);
      for q in Q do R[q]:= x/q od;
      Agenda:= Agenda minus Q;
    fi;
  od;
od:
seq(R[i],i=0..80); # Robert Israel, Apr 18 2025

Mathematica

a[n_] := For[k = 1, True, k++, id = IntegerDigits[k*n]; If[AllTrue[id, EvenQ], rid = Reverse[id]; If[id == rid || (id //. {d__, 0} :> {d}) == (rid //. {0, d__} :> {d}), Return[k]]]]; a[0] = 1; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Apr 01 2016 *)
skpal[n_]:=Module[{k=1},While[Count[IntegerDigits[k n],?OddQ]>0 || (!PalindromeQ[(k n)/10^IntegerExponent[n k]]),k++];k]; Array[skpal,70,0] (* _Harvey P. Dale, Dec 19 2021 *)

Extensions

More terms from Klaus Brockhaus, Jun 27 2001