A008919 Numbers k such that k written backwards is a nontrivial multiple of k.

1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• nice

Comments

There are 2*Fibonacci(floor((n-2)/2)) terms with n digits (this is A214927 or essentially twice A103609). - N. J. A. Sloane, Mar 20 2013

All terms are made of "symmetric" concatenations of 1089 and/or 2178, with an arbitrary numbers of 9's inserted in the middle of these and 0's inserted between them. See A031877 for the reversals and further information: union of the two, sequences "made of" 1089 or 2178 only. - M. F. Hasler, Jun 23 2019

Also: 99 times A061852: numbers that are palindromic, have only digits in {0, 1} or in {0, 2}, and no isolated ("single") digit. - M. F. Hasler, Oct 17 2022

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.
• Gardiner, Anthony, and A. D. Gardiner. Discovering mathematics: The art of investigation. Oxford University Press, 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").
• D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 400 terms from Vincenzo Librandi)
• Yannis Almirantis and Wentian Li, Iterative Digital Reversion: a simple algorithm deploying a complex phenomenology related to the '1089 effect', ResearchGate, 2024. See p. 17.
• 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, Derived Palintiple Families and Their Palinomials, 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.
• Benjamin V. Holt, Finding permultiples of a known base and multiplier, Integers (2025) Vol. 25, Art. No. A13. See p. 23, also arXiv:2411.10859 [math.CO], 2024.
• Benjamin V. Holt, On the Reflective Symmetry of the Mother Graph, arXiv:2504.17158 [math.CO], 2025. See p. 28.
• Benjamin V. Holt, A Multigraph Characterization of Permutiple Strings, arXiv:2505.11414 [math.CO], 2025. See p. 17.
• 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.
• Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210.
• 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.
• N. J. A. Sloane, 2178 And All That [Local copy]
• R. Webster and G. Williams, On the Trail of Reverse Divisors: 1089 and All that Follow, Mathematical Spectrum, Applied Probability Trust, Sheffield, Vol. 45, No. 3, 2012/2013, pp. 96-102.
• Eric Weisstein's World of Mathematics, Reversal

Crossrefs

Cf. A001232 (9k = R(k)), A004086 (R(n): reverse), A008918 (4k = R(k)), A214927, A103609 (Fibonacci([n/2])). Reversals are in A031877.

Programs

Haskell

a008919 n = a008919_list !! (n-1)
a008919_list = [x | x <- [1..],
                    let (x',m) = divMod (a004086 x) x, m == 0, x' > 1]
-- Reinhard Zumkeller, Feb 03 2012

Mathematica

Reap[ Do[ If[ Reverse[ IntegerDigits[n]] == IntegerDigits[4*n], Print[n]; Sow[n]]; If[ Reverse[ IntegerDigits[n + 11]] == IntegerDigits[9*(n + 11)], Print[n + 11]; Sow[n + 11]], {n, 78, 2*10^10, 100}]][[2, 1]] (* Jean-François Alcover, Jun 19 2012, after David W. Wilson, assuming n congruent to 78 or 89 mod 100 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; Sort[ Flatten[ {99#, 198#}&/@Flatten[Table[FromDigits/@Select[Tuples[ {0,1},n], okQ],{n,10}]]]] (* Harvey P. Dale, Jul 03 2013 *)

PARI

is_A008919(n,r=A004086(n))={n>r && n%r==0} \\ M. F. Hasler, Jun 23 2019

Formula

If reverse(n) = k*n in base 10, then k = 1, 4 or 9 [Klosinski and Smolarski]. Hence A008919 is the union of A001232 and A008918. - David W. Wilson

a(n) = 99*A061852(n). - M. F. Hasler, Oct 17 2022

Extensions

Corrected and extended by David W. Wilson Aug 15 1996, Dec 15 1997