|
%I #77 Apr 10 2020 07:44:11
%S 8712,9801,87912,98901,879912,989901,8799912,9899901,87128712,
%T 87999912,98019801,98999901,871208712,879999912,980109801,989999901,
%U 8712008712,8791287912,8799999912,9801009801,9890198901,9899999901,87120008712,87912087912,87999999912
%N Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.
%C The terms of this sequence are sometimes called palintiples.
%C All terms are of the form 87...12 = 4*21...78 or 98...01 = 9*10...89. [This was proved by Hoey, 1992. - _N. J. A. Sloane_, Oct 19 2014] More precisely, they are obtained from concatenated copies of either 8712 or 9801, with 9's inserted "in the middle of" these and/or 0's inserted between the copies these, in a symmetrical way. A008919 lists the reversals, but not in the same order, e.g., R(a(2)) < R(a(1)). - _M. F. Hasler_, Aug 18 2014
%C There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - _Ray Chandler_, Oct 11 2017
%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 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.").
%H Ray Chandler, <a href="/A031877/b031877.txt">Table of n, a(n) for n = 1..10000</a>
%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">A Determination of Symmetric Palintiples</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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Reversal.html">Reversal.</a>
%F a(n) = A004086(a(n))*[9/(a(n)%10)], where [...]=9 if a(n) ends in "1" and [...]=4 if a(n) ends in "2". - _M. F. Hasler_, Aug 18 2014
%t fQ[n_] := Block[{id = IntegerDigits@n}, Mod[n, FromDigits@ Reverse@id] == 0 && n != FromDigits@ Reverse@ id && Mod[n, 10] > 0]; k = 1; lst = {}; While[k < 10^9, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* _Robert G. Wilson v_, Jun 11 2010 *)
%t okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[Flatten[ {(4*198)#,(9*99)#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,12}]]]] (* _Harvey P. Dale_, Jul 03 2013 *)
%o (Haskell)
%o a031877_list = [x | x <- [1..], x `mod` 10 > 0,
%o let x' = a004086 x, x' /= x && x `mod` x' == 0]
%o -- _Reinhard Zumkeller_, Jul 15 2013
%o (PARI) is_A031877(n)={n%10 && n%A004086(n)==0 && n>A004086(n)} \\ _M. F. Hasler_, Aug 18 2014
%o (Python)
%o A031877 = []
%o for n in range(1,10**7):
%o if n % 10:
%o s1 = str(n)
%o s2 = s1[::-1]
%o if s1 != s2 and not n % int(s2):
%o A031877.append(n) # _Chai Wah Wu_, Sep 05 2014
%Y See A008919 for reversals (this is the main entry for the problem).
%Y Cf. A169824, A214927, A103609.
%Y Union of A222814 and A222815.
%Y Subsequence of A118959.
%K nonn,base
%O 1,1
%A _Eric W. Weisstein_
%E More terms from _Jud McCranie_, Aug 15 2001
%E More terms from _Sam Mathers_, Aug 18 2014
| 