A061852 -id:A061852 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

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

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

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

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 *)

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

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

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

A061851 Digital representation of n contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's.

Original entry on oeis.org

11, 111, 1111, 11111, 110011, 111111, 1100011, 1111111, 11000011, 11100111, 11111111, 110000011, 111000111, 111111111, 1100000011, 1100110011, 1110000111, 1111001111, 1111111111, 11000000011, 11001110011, 11100000111, 11110001111, 11111111111, 110000000011, 110001100011

Offset: 1

Henry Bottomley, May 10 2001

The terms can be constructed by gluing together terms from A355280 with their reversal as follows: The terms with odd length L = 2k-1 are given from the k-digit terms of A355280 by replacing the last digit with the reversal of the term. (Equivalently, concatenate with the reversal and delete one of the middle digits.) Terms with an even number L = 2k of digits are given as concatenation(m, reverse(m)) = m*10^L(m) + A004086(m) where m runs over the k-digit terms from A355280, and the (k-1)-digit terms with the 1's complement of the last digit appended. This explains the formula given in CROSSREFS for the number of terms of given length. - M. F. Hasler, Oct 17 2022

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 608 terms from N. J. A. Sloane)
N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Number of terms with k digits is Fibonacci(floor(k/2)) = A000045(A004526(k)).
Union of this sequence and twice this sequence is A061852 and 99 times that is A008919.
Cf. A214927, A001232, A222814, A008918, A008919, A222813 (terms converted to decimal).
Cf. A355280 (palindromic binary numbers with no digit run of length < 2).

concat(apply( {A061851_row(n)=[fromdigits(binary(m))|m<-A222813_row(n)]}, [1..11])) \\ M. F. Hasler, Oct 17 2022

def A061851_row(n): return [] if n < 2 else [10**n//9] if n < 6 else [
    m*10**(n//2) + A004086(m//10) for m in A355280_row(n//2+1)] if n&1 else [
    m*10**(n//2) + A004086(m) for m in sorted(A355280_row(n//2)+
                        [x*10+1-x%10 for x in A355280_row(n//2-1)])]
# M. F. Hasler, Oct 17 2022

a(n) = A001232(n)/99 = A008918(n)/198.

a(n) = A007088(A222813(n)), where A007088 = write in binary. - M. F. Hasler, Oct 06 2022

cp's OEIS Frontend

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

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