cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A061816 Obtain m by omitting trailing zeros from n (cf. A004151); a(n) = smallest multiple k*m which is a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 252, 494, 252, 525, 272, 272, 252, 171, 2, 252, 22, 161, 696, 525, 494, 999, 252, 232, 3, 434, 2112, 33, 272, 525, 252, 111, 494, 585, 4, 656, 252, 989, 44, 585, 414, 141, 2112, 343, 5, 969, 676, 212, 27972, 55, 616, 171, 232
Offset: 0

Views

Author

Klaus Brockhaus, Jun 25 2001

Keywords

Comments

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see links).
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

Examples

			For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 3. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 252.

Links

Crossrefs

Cf. A050782, A062293, A061915, A061916. Values of k are given in A061906.

Programs

  • ARIBAS
    stop := 200000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(mp," "); else write(-1," "); end; end;

A062279 Smallest multiple k*n of n which is a palindrome or becomes a palindrome when 0's are added on the left (e.g. 10 becomes 010 which is a palindrome).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 60, 494, 70, 30, 80, 272, 90, 171, 20, 252, 22, 161, 600, 50, 494, 999, 252, 232, 30, 434, 800, 33, 272, 70, 252, 111, 494, 585, 40, 656, 252, 989, 44, 90, 414, 141, 2112, 343, 50, 969, 676, 212, 9990, 55, 616, 171, 232, 767
Offset: 0

Views

Author

Amarnath Murthy, Jun 17 2001

Keywords

Comments

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).

Examples

			a(13) = 494 is the smallest multiple of 13 which is a palindrome.

Links

Crossrefs

Cf. A050782, A062293. Values of k are given in A061674.
Cf. A141709.

Programs

  • ARIBAS
    : maxarg := 60; stop := 200000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := m <> int_reverse(m) then inc(k); end; end; if k < stop then write(n*k," "); else write(-1," "); end; end;
  • Haskell
    a062279 0 = 0
a062279 n = until ((== 1) . a136522 . a004151) (+ n) n
-- Reinhard Zumkeller, May 06 2013

Formula

A136522(A004151(a(n))) = 1. - Reinhard Zumkeller, May 06 2013

Extensions

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Klaus Brockhaus, Jun 18 2001

A061906 Obtain m by omitting trailing zeros from n; a(n) = smallest k such that k*m is a palindrome.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 38, 18, 35, 17, 16, 14, 9, 1, 12, 1, 7, 29, 21, 19, 37, 9, 8, 1, 14, 66, 1, 8, 15, 7, 3, 13, 15, 1, 16, 6, 23, 1, 13, 9, 3, 44, 7, 1, 19, 13, 4, 518, 1, 11, 3, 4, 13, 1, 442, 7, 4, 33, 9, 1, 11, 4, 6, 1, 845, 88, 4, 3, 7, 287, 1, 11, 6, 1, 12345679, 8
Offset: 0

Views

Author

Klaus Brockhaus, Jun 25 2001

Keywords

Comments

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).
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

Examples

			For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 1. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 21.

Links

Crossrefs

Cf. A050782, A062293, A061915, A061916, A061816. Values of k*m are given in A061906.

Programs

  • ARIBAS
    stop := 20000000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;
  • Mathematica
    skp[n_]:=Module[{m=n/10^IntegerExponent[n,10],k=1},While[!PalindromeQ[k*m],k++];k]; Array[ skp,90,0] (* Harvey P. Dale, Jul 04 2024 *)
  • Python
    from _future_ import division
def palgen(l, b=10): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n, n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = divmod(k, b)
                    m = b*m + r
                yield y*n + b*m + k
            for y in range(n, n2):
                k, m = y, 0
                while k >= b:
                    k, r = divmod(k, b)
                    m = b*m + r
                yield y*n2 + b*m + k
def A050782(n, l=10):
    if n % 10:
        x = palgen(l)
        next(x)  # replace with x.next() in Python 2.x
        for i in x:
            q, r = divmod(i, n)
            if not r:
                return q
        else:
            return 'search limit reached.'
    else:
        return 0
def A061906(n, l=10):
    return A050782(int(str(n).rstrip('0')),l) if n > 0 else 1
# Chai Wah Wu, Dec 30 2014

A061674 Smallest k such that k*n is a palindrome or becomes a palindrome when 0's are added on the left.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 38, 5, 2, 5, 16, 5, 9, 1, 12, 1, 7, 25, 2, 19, 37, 9, 8, 1, 14, 25, 1, 8, 2, 7, 3, 13, 15, 1, 16, 6, 23, 1, 2, 9, 3, 44, 7, 1, 19, 13, 4, 185, 1, 11, 3, 4, 13, 1, 442, 7, 4, 33, 9, 1, 11, 4, 6, 1, 845, 35, 4, 3, 4, 65, 1, 11, 6, 1, 12345679, 8, 9, 3
Offset: 0

Views

Author

Amarnath Murthy, Jun 17 2001

Keywords

Comments

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).

Examples

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

Links

Crossrefs

Cf. A050782, A062293. Values of k*n are given in A062279.

Programs

  • ARIBAS
    stop := 50000000; for n := 0 to 100 do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := m <> int_reverse(m) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;
  • Haskell
    a061674 n = until ((== 1) . a136522 . a004151 . (* n)) (+ 1) 1
-- Reinhard Zumkeller, Jul 20 2012
  • Mathematica
    rz[n_]:=Module[{idn=IntegerDigits[n]},While[Last[idn]==0,idn=Most[idn]];idn]; k[n_]:=Module[{k=1,p},p=k*n;While[rz[p]!=Reverse[rz[p]],k++;p=k*n];k]; Join[ {1},Array[k,90]] (* Harvey P. Dale, Mar 06 2013 *)

A061915 Obtain m by omitting trailing zeros from n; a(n) = smallest multiple k*m which is a palindrome with even digits, or -1 if no such multiple exists.

Original entry on oeis.org

0, 2, 2, 6, 4, -1, 6, 686, 8, 666, 2, 22, 444, 2002, 686, -1, 464, 646, 666, 646, 2, 6006, 22, 828, 888, -1, 2002, 8886888, 868, 464, 6, 868, 4224, 66, 646, -1, 828, 222, 646, 6006, 4, 22222, 6006, 68886, 44, -1, 828, 282, 4224, 686, -1, 42024, 4004, 424, 8886888, -1, 8008, 68286, 464, 68086, 6
Offset: 0

Views

Author

Klaus Brockhaus, Jun 25 2001

Keywords

Comments

a(n) = -1 if and only if m ends with the digit 5.

Examples

			For n = 30 we have m = 3; 3*2 = 6 is a palindrome with even digits, so a(30) = 6.

Links

Crossrefs

Cf. A050782, A062293, A061816, A061906. Values of k are given in A061916.

Programs

  • ARIBAS
    stop := 500000; for n := 0 to 60 do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := not all_even(mp) or mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(mp," "); else write(-1," "); end; end;

A061916 Obtain m by omitting trailing zeros from n; a(n) = smallest k such that k*m is a palindrome with even digits, or -1 if no such multiple exists.

Original entry on oeis.org

1, 2, 1, 2, 1, -1, 1, 98, 1, 74, 2, 2, 37, 154, 49, -1, 29, 38, 37, 34, 1, 286, 1, 36, 37, -1, 77, 329144, 31, 16, 2, 28, 132, 2, 19, -1, 23, 6, 17, 154, 1, 542, 143, 1602, 1, -1, 18, 6, 88, 14, -1, 824, 77, 8, 164572, -1, 143, 1198, 8, 1154, 1, 1126, 14, 962, 66, -1, 1, 998, 121, 12, 98, 65984, 592, 274, 3, -1, 529, 26, 77, 358
Offset: 0

Views

Author

Klaus Brockhaus, Jun 25 2001

Keywords

Comments

a(n) = -1 if and only if m ends with the digit 5.

Examples

			For n = 30 we have m = 3; 3*2 = 6 is a palindrome with even digits, so a(30) = 2.

Links

Crossrefs

Cf. A050782, A062293, A061816, A061906. Values of k*m are given in A061915.

Programs

  • ARIBAS
    stop := 500000; for n := 0 to 80 do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := not all_even(mp) or mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;

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.

Original entry on oeis.org

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

Views

Author

Amarnath Murthy, Jun 17 2001

Keywords

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

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
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.