

A031877


Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.


18



8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, 871208712, 879999912, 980109801, 989999901, 8712008712, 8791287912, 8799999912, 9801009801, 9890198901, 9899999901, 87120008712, 87912087912, 87999999912
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OFFSET

1,1


COMMENTS

The terms of this sequence are sometimes called palintiples.
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
There are 2*Fibonacci(floor((n2)/2)) terms with n digits (this is A214927 or essentially twice A103609).  Ray Chandler, Oct 11 2017


REFERENCES

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays (1939, page 13); 13th ed. New York: Dover, pp. 1415, 1987.
G. H. Hardy, A Mathematician's Apology (Cambridge Univ. Press, 1940, reprinted 2000), pp. 104105 (describes this problem as having "nothing in [it] which appeals much to a mathematician.").


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000
Martin Beech, A Computer Conjecture of a NonSerious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 5051.
Patrick De Geest, Palindromic Products of Integers and their Reversals
D. J. Hoey, Palintiples
D. J. Hoey, Palintiples [Cached copy]
Benjamin V. Holt, Some General Results and Open Questions on Palintiple Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 14, Paper A42, 2014.
Benjamin V. Holt, A Determination of Symmetric Palintiples, arXiv:1410.2356 [math.NT], 2014.
Benjamin V. Holt, Families of Asymmetric Palintiples Constructed from Symmetric and ShiftedSymmetric Palintiples, arXiv:1412.0231 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106 [math.NT], 2014.
Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129132.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99120.
Eric Weisstein's World of Mathematics, Reversal.


FORMULA

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


MATHEMATICA

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 *)
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 *)


PROG

(Haskell)
a031877_list = [x  x < [1..], x `mod` 10 > 0,
let x' = a004086 x, x' /= x && x `mod` x' == 0]
 Reinhard Zumkeller, Jul 15 2013
(PARI) is_A031877(n)={n%10 && n%A004086(n)==0 && n>A004086(n)} \\ M. F. Hasler, Aug 18 2014
(Python)
A031877 = []
for n in range(1, 10**7):
....if n % 10:
........s1 = str(n)
........s2 = s1[::1]
........if s1 != s2 and not n % int(s2):
............A031877.append(n) # Chai Wah Wu, Sep 05 2014


CROSSREFS

See A008919 for reversals (this is the main entry for the problem).
Cf. A169824, A214927, A103609.
Union of A222814 and A222815.
Subsequence of A118959.
Sequence in context: A237874 A252646 A170796 * A222815 A233679 A035909
Adjacent sequences: A031874 A031875 A031876 * A031878 A031879 A031880


KEYWORD

nonn,base


AUTHOR

Eric W. Weisstein


EXTENSIONS

More terms from Jud McCranie, Aug 15 2001
More terms from Sam Mathers, Aug 18 2014


STATUS

approved



