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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 (list; graph; refs; listen; history; text; internal format)
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((n-2)/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. 14-15, 1987.
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.").
LINKS
Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
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.
L. H. Kendrick, Young Graphs: 1089 et al, arXiv:1410.0106 [math.NT], 2014.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
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).
Union of A222814 and A222815.
Subsequence of A118959.
Sequence in context: A252646 A170796 A035909 * A222815 A233679 A266578
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Aug 15 2001
More terms from Sam Mathers, Aug 18 2014
STATUS
approved

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Last modified March 3 00:25 EST 2024. Contains 370499 sequences. (Running on oeis4.)