

A031877


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


21



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

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]
(Python)
for n in range(1, 10**7):
if n % 10:
s1 = str(n)
s2 = s1[::1]
if s1 != s2 and not n % int(s2):


CROSSREFS

See A008919 for reversals (this is the main entry for the problem).


KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



