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%I #40 Feb 07 2022 03:41:32
%S 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,
%T 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,
%U 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
%N Smallest positive multiplier m such that m*n is palindromic (or zero if no such m exists).
%C Multiples of 81 require the largest multipliers.
%C From _Jon E. Schoenfield_, Jan 15 2015: (Start)
%C 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,
%C 755 are in [1..9],
%C 1760 are in [10..99],
%C 3439 are in [100..999],
%C 2180 are in [1000..9999],
%C 708 are in [10000..99999],
%C 36 are in [100000..999999],
%C 6 are in [1000000..9999999],
%C 2 are in [10000000..99999999],
%C 2 are in [100000000..999999999],
%C and 1 (the largest) is a(8891) = 8546948927,
%C but the smallest of the 111 terms whose indices are multiples of 81 is a(2997)=333667. (End)
%C 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
%H Giovanni Resta, <a href="/A050782/b050782.txt">Table of n, a(n) for n = 0..10000</a> (first 8181 terms from Chai Wah Wu)
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/index.html">World!Of Numbers</a>
%e E.g., a(81) -> 81 * 12345679 = 999999999 and a palindrome.
%p digrev:= proc(n) local L,d,i;
%p L:= convert(n,base,10);
%p d:= nops(L);
%p add(L[i]*10^(d-i),i=1..d);
%p end proc:
%p f:= proc(n)
%p local d,d2,x,t,y;
%p if n mod 10 = 0 then return 0 fi;
%p if n < 10 then return 1 fi;
%p for d from 2 do
%p if d::even then
%p d2:= d/2;
%p for x from 10^(d2-1) to 10^d2-1 do
%p t:= x*10^d2 + digrev(x);
%p if t mod n = 0 then return(t/n) fi;
%p od
%p else
%p d2:= (d-1)/2;
%p for x from 10^(d2-1) to 10^d2-1 do
%p for y from 0 to 9 do
%p t:= x*10^(d2+1)+y*10^d2+digrev(x);
%p if t mod n = 0 then return(t/n) fi;
%p od
%p od
%p fi
%p od;
%p end proc:
%p seq(f(n),n=0 .. 100); # _Robert Israel_, Jan 15 2015
%t 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 *)
%o (Python)
%o from __future__ import division
%o def palgen(l,b=10): # generator of palindromes in base b of length <= 2*l
%o if l > 0:
%o yield 0
%o for x in range(1,l+1):
%o n = b**(x-1)
%o n2 = n*b
%o for y in range(n,n2):
%o k, m = y//b, 0
%o while k >= b:
%o k, r = divmod(k,b)
%o m = b*m + r
%o yield y*n + b*m + k
%o for y in range(n,n2):
%o k, m = y, 0
%o while k >= b:
%o k, r = divmod(k,b)
%o m = b*m + r
%o yield y*n2 + b*m + k
%o def A050782(n, l=10):
%o if n % 10:
%o x = palgen(l)
%o next(x) # replace with x.next() in Python 2.x
%o for i in x:
%o q, r = divmod(i, n)
%o if not r:
%o return q
%o else:
%o return 'search limit reached.'
%o else:
%o return 0 # _Chai Wah Wu_, Dec 30 2014
%Y Cf. A002113, A020485, A050810.
%K nonn,base,nice
%O 0,13
%A _Patrick De Geest_, Oct 15 1999
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