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
Ray Chandler, Table of n, a(n) for n = 1..10000
Martin Beech, A Computer Conjecture of a Non-Serious Theorem, Mathematical Gazette, 74 (No. 467, March 1990), 50-51.
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 Shifted-Symmetric Palintiples, arXiv:1412.0231 [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
(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
KEYWORD
nonn,base
AUTHOR
EXTENSIONS
More terms from Jud McCranie, Aug 15 2001
More terms from Sam Mathers, Aug 18 2014
STATUS
approved