%I #27 Aug 11 2024 14:41:33

%S 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,

%T 329144,31,16,2,28,25,2,19,196,23,6,17,154,1,542,143,1602,1,148,18,6,

%U 88,14,4,824,77,8,164572,4,143,1198,8,1154,1,1126,14,962,66,308,1,998

%N 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.

%C 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

%C a(81), if it exists, is greater than 5 million. - _Harvey P. Dale_, Dec 19 2021

%H Reinhard Zumkeller, <a href="/A061797/b061797.txt">Table of n, a(n) for n = 0..80</a>

%H P. De Geest, <a href="https://www.worldofnumbers.com/em36.htm">Smallest multipliers to make a number palindromic</a>.

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

%t 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 *)

%t 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 *)

%o (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;

%o (Haskell)

%o a061797 0 = 1

%o a061797 n = head [k | k <- [1..], let x = k * n,

%o all (`elem` "02468") $ show x, a136522 (a004151 x) == 1]

%o -- _Reinhard Zumkeller_, Feb 01 2012

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

%Y Cf. A014263, A136522, A004151.

%K nonn,base,easy,nice

%O 0,2

%A _Amarnath Murthy_, Jun 17 2001

%E More terms from _Klaus Brockhaus_, Jun 27 2001