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%I #100 Feb 16 2024 10:09:09
%S 1089,2178,10989,21978,109989,219978,1099989,2199978,10891089,
%T 10999989,21782178,21999978,108901089,109999989,217802178,219999978,
%U 1089001089,1098910989,1099999989,2178002178,2197821978,2199999978,10890001089
%N Numbers k such that k written backwards is a nontrivial multiple of k.
%C There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - _N. J. A. Sloane_, Mar 20 2013
%C All terms are made of "symmetric" concatenations of 1089 and/or 2178, with an arbitrary numbers of 9's inserted in the middle of these and 0's inserted between them. See A031877 for the reversals and further information: union of the two, sequences "made of" 1089 or 2178 only. - _M. F. Hasler_, Jun 23 2019
%C Also: 99 times A061852: numbers that are palindromic, have only digits in {0, 1} or in {0, 2}, and no isolated ("single") digit. - _M. F. Hasler_, Oct 17 2022
%D W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 14-15, 1987.
%D Gardiner, Anthony, and A. D. Gardiner. Discovering mathematics: The art of investigation. Oxford University Press, 1987.
%D G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104-105 (describes this problem as having "nothing in [it] which appeals much to a mathematician").
%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.
%H Ray Chandler, <a href="/A008919/b008919.txt">Table of n, a(n) for n = 1..10000</a> (first 400 terms from Vincenzo Librandi)
%H Martin Beech, <a href="http://www.jstor.org/stable/3618854">A Computer Conjecture of a Non-Serious Theorem</a>, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/reversal.htm">Palindromic Products of Integers and their Reversals</a>
%H D. J. Hoey, <a href="http://djm.cc/rpa-output/arithmetic/digits/palintiples.s">Palintiples</a>
%H D. J. Hoey, <a href="/A008919/a008919.txt">Palintiples</a> [Cached copy]
%H Benjamin V. Holt, <a href="http://www.emis.de/journals/INTEGERS/papers/o42/o42.Abstract.html">Some General Results and Open Questions on Palintiple Numbers</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.
%H Benjamin V. Holt, <a href="http://arxiv.org/abs/1410.2356">Derived Palintiple Families and Their Palinomials</a>, arXiv:1410.2356 [math.NT], 2014.
%H Benjamin V. Holt, <a href="http://arxiv.org/abs/1412.0231">Families of Asymmetric Palintiples Constructed from Symmetric and Shifted-Symmetric Palintiples</a>, arXiv:1412.0231 [math.NT], 2014.
%H L. H. Kendrick, <a href="http://arxiv.org/abs/1410.0106">Young Graphs: 1089 et al.</a>, arXiv:1410.0106 [math.NT], 2014.
%H L. H. Kendrick, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kendrick/ken1.html">Young Graphs: 1089 et al.</a>, J. Int. Seq. 18 (2015) 15.9.7.
%H Leonard F. Klosinski and Dennis C. Smolarski, <a href="http://www.jstor.org/stable/2688542">On the Reversing of Digits</a>, Math. Mag., 42 (1969), 208-210.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/mm005281.pdf">Digit Reversal Without Apology</a>, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1307.0453">2178 And All That</a>, Fib. Quart., 52 (2014), 99-120.
%H R. Webster and G. Williams, <a href="http://users.mct.open.ac.uk/gw3285/publications/reverse-divisors.pdf">On the Trail of Reverse Divisors: 1089 and All that Follow</a>, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96-102.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Reversal.html">Reversal</a>
%F If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - _David W. Wilson_
%F a(n) = 99*A061852(n). - _M. F. Hasler_, Oct 17 2022
%t Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* _Jean-François Alcover_, Jun 19 2012, after _David W. Wilson_, assuming n congruent to 78 or 89 mod 100 *)
%t okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,10}]]]] (* _Harvey P. Dale_, Jul 03 2013 *)
%o (Haskell)
%o a008919 n = a008919_list !! (n-1)
%o a008919_list = [x | x <- [1..],
%o let (x',m) = divMod (a004086 x) x, m == 0, x' > 1]
%o -- _Reinhard Zumkeller_, Feb 03 2012
%o (PARI) is_A008919(n,r=A004086(n))={n>r && n%r==0} \\ _M. F. Hasler_, Jun 23 2019
%Y Cf. A001232 (9k = R(k)), A004086 (R(n): reverse), A008918 (4k = R(k)), A214927, A103609 (Fibonacci([n/2])). Reversals are in A031877.
%K nonn,base,nice
%O 1,1
%A _N. J. A. Sloane_
%E Corrected and extended by _David W. Wilson_ Aug 15 1996, Dec 15 1997
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