A062279 -id:A062279 - cp's OEIS frontend

A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

0, 2, 2, 6, 4, 20, 6, 686, 8, 666, 20, 22, 60, 2002, 686, 60, 80, 646, 666, 646, 20, 6006, 22, 828, 600, 200, 2002, 8886888, 868, 464, 60, 868, 800, 66, 646, 6860, 828, 222, 646, 6006, 40, 22222, 6006, 68886, 44, 6660, 828, 282, 4224, 686, 200, 42024, 4004, 424, 8886888, 220, 8008, 68286, 464, 68086, 60

Offset: 0

Amarnath Murthy, Jun 18 2001

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(7) = 686 as 686 = 98*7 is the smallest palindrome multiple of 7 with even digits.

Cf. A062279. Values of k are given in A061797.

Cf. A014263, A136522, A004151.

stop := 500000; for n := 0 to 60 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(n*k," "); else write(-1," "); end; end;

a062293 0 = 0
a062293 n = head [x | x <- map (* n) [1..],
                 all (`elem` "02468") $ show x, a136522 (a004151 x) == 1]
-- Reinhard Zumkeller, Feb 01 2012

Corrected and extended by Klaus Brockhaus, Jun 21 2001

A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 12, 65, 14, 15, 16, 17, 18, 513, 20, 21, 66, 2047, 24, 325, 130, 27, 28, 1421, 30, 31, 32, 33, 34, 455, 36, 2553, 1026, 195, 40, 1025, 42, 129, 132, 45, 4094, 4841, 48, 1421, 650, 51, 260, 3339, 54, 165, 56, 513, 2842, 6077, 60, 427, 62

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2, unless leading zeros are considered (or, equivalently, resp. more precisely, trailing zeros are discarded). This is done in this version of A141708, which therefore does not need to be restricted to odd n as it has been done for A141707 and A141708.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050782, A141707-A141708, A062279, A203070, A000265, A178225.

a141709 n = until ((== 1) . a178225 . a000265) (+ n) n
-- Reinhard Zumkeller, Nov 06 2012

notpalbinQ[i_]:=Module[{id=IntegerDigits[i,2]},While[Last[id]==0,id=Most[id]];id!= Reverse[id]]; lm[n_]:=Module[{k=1},While[notpalbinQ[k n],k++];k n]; Array[lm,70] (* Harvey P. Dale, Dec 28 2011 *)

A141709(n)=forstep(k=n,10^9,n,vecextract(t=binary(k>>valuation(k,2)),"-1..1")-t || return(k))

A178225(A000265(a(n))) = 1. - Reinhard Zumkeller, Nov 06 2012

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

Amarnath Murthy, Jun 17 2001

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

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..2500
Patrick De Geest, Smallest multipliers to make a number palindromic.

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

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;

a061674 n = until ((== 1) . a136522 . a004151 . (* n)) (+ 1) 1
-- Reinhard Zumkeller, Jul 20 2012

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 *)

A141708 Least positive multiple of 2n-1 which is palindromic in base 2.

Original entry on oeis.org

1, 3, 5, 7, 9, 33, 65, 15, 17, 513, 21, 2047, 325, 27, 1421, 31, 33, 455, 2553, 195, 1025, 129, 45, 4841, 1421, 51, 3339, 165, 513, 6077, 427, 63, 65, 1273, 2553, 10437, 73, 975, 231, 1501, 891, 3735, 85, 3219, 2047, 273, 93, 2565, 5917, 99, 23533, 4841, 1365, 107

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2 (unless leading zeros are considered), that's why we search for odd numbers 2n-1 their smallest multiple k(2n-1) which is palindromic in base 2. Obviously this must always be odd.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050782, A141707, A062279.

a141708 n = a141707 n * (2 * n - 1) -- Reinhard Zumkeller, Apr 20 2015

pal2[n_]:=Module[{k=1},While[IntegerDigits[k n,2] != Reverse[ IntegerDigits[ k n,2]],k++];k n]; pal2/@Range[1,121,2] (* Harvey P. Dale, Feb 29 2012 *)

A141708(n,L=10^9)={ n=2*n-1; forstep(k=1,L,2, binary(k*n)-vecextract(binary(k*n),"-1..1") || return(k*n))}

def binpal(n): b = bin(n)[2:]; return b == b[::-1]
def a(n):
    m = 2*n - 1
    km = m
    while not binpal(km): km += m
    return km
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Mar 20 2022

a(n) = (2n-1)*A141707(n).

cp's OEIS Frontend

A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

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A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

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

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Keywords

Comments

Examples

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Crossrefs

Programs

ARIBAS

Haskell

Mathematica

A141708 Least positive multiple of 2n-1 which is palindromic in base 2.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula