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A050782 Smallest positive multiplier m such that m*n is palindromic (or zero if no such m exists). 19
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.
From Jon E. Schoenfield, Jan 15 2015: (Start)
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. (End)
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
Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 8181 terms from Chai Wah Wu)
Patrick 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
Sequence in context: A083567 A109211 A224701 * A061906 A139768 A307278
KEYWORD
nonn,base,nice
AUTHOR
Patrick De Geest, Oct 15 1999
STATUS
approved

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Last modified February 26 13:41 EST 2024. Contains 370352 sequences. (Running on oeis4.)