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A050782 Smallest multiplier m (>0) such that mn is palindromic (or zero if no such m exists). 15
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 21, 38, 18, 35, 17, 16, 14, 9, 0, 12, 1, 7, 29, 21, 19, 37, 9, 8, 0, 14, 66, 1, 8, 15, 7, 3, 13, 15, 0, 16, 6, 23, 1, 13, 9, 3, 44, 7, 0, 19, 13, 4, 518, 1, 11, 3, 4, 13, 0, 442, 7, 4, 33, 9, 1, 11, 4, 6, 0, 845, 88, 4, 3, 7, 287, 1, 11, 6, 0, 12345679, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Multiples of 81 require the largest multipliers.

In general, a(n) is large when n is a multiple of 81. E.g., for n in [1..10000], of the 9000 terms where a(n)>0, 111 are at indices n that are multiples of 81; of the remaining 8889 terms,

     755 are in [1..9],

    1760 are in [10..99],

    3439 are in [100..999],

    2180 are in [1000..9999],

     708 are in [10000..99999],

      36 are in [100000..999999],

       6 are in [1000000..9999999],

       2 are in [10000000..99999999],

       2 are in [100000000..999999999],

and 1 (the largest) is a(8891) = 8546948927,

but the smallest of the 111 terms whose indices are multiples of 81 is a(2997)=333667. - Jon E. Schoenfield, Jan 15 2015

a(n) = 0 iff 10 | n. a(n) = 1 iff n is a palindrome. If k | a(n) then a(k*n) = a(n)/k. - Robert Israel, Jan 15 2015

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..8180

P. De Geest, World!Of Numbers

EXAMPLE

E.g., a(81) -> 81 * 12345679 = 999999999 and a palindrome.

MAPLE

digrev:= proc(n) local L, d, i;

  L:= convert(n, base, 10);

  d:= nops(L);

  add(L[i]*10^(d-i), i=1..d);

end proc:

f:= proc(n)

local d, d2, x, t, y;

if n mod 10 = 0 then return 0 fi;

if n < 10 then return 1 fi;

for d from 2 do

  if d::even then

    d2:= d/2;

    for x from 10^(d2-1) to 10^d2-1 do

       t:= x*10^d2 + digrev(x);

       if t mod n = 0 then return(t/n) fi;

    od

  else

    d2:= (d-1)/2;

    for x from 10^(d2-1) to 10^d2-1 do

      for y from 0 to 9 do

        t:= x*10^(d2+1)+y*10^d2+digrev(x);

        if t mod n = 0 then return(t/n) fi;

      od

    od

  fi

od;

end proc:

seq(f(n), n=0 .. 100); # Robert Israel, Jan 15 2015

MATHEMATICA

t={0}; Do[i=1; If[IntegerQ[n/10], y=0, While[Reverse[x=IntegerDigits[i*n]]!=x, i++]; y=i]; AppendTo[t, y], {n, 80}]; t (* Jayanta Basu, Jun 01 2013 *)

PROG

(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 # Chai Wah Wu, Dec 30 2014

CROSSREFS

Cf. A002113, A050810.

Sequence in context: A083567 A109211 A224701 * A061906 A139768 A176071

Adjacent sequences:  A050779 A050780 A050781 * A050783 A050784 A050785

KEYWORD

nonn,base,nice

AUTHOR

Patrick De Geest, Oct 15 1999

STATUS

approved

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Last modified March 26 01:08 EDT 2019. Contains 321479 sequences. (Running on oeis4.)