A031877 -id:A031877 - cp's OEIS frontend

A214927 Number of n-digit numbers N that do not end with 0 and are such that the reversal of N divides N but is different from N.

0, 0, 0, 2, 2, 2, 2, 4, 4, 6, 6, 10, 10, 16, 16, 26, 26, 42, 42, 68, 68, 110, 110, 178, 178, 288, 288, 466, 466, 754, 754, 1220, 1220, 1974, 1974, 3194, 3194, 5168, 5168, 8362, 8362, 13530, 13530, 21892, 21892, 35422, 35422, 57314, 57314, 92736, 92736, 150050, 150050, 242786, 242786, 392836, 392836, 635622, 635622

Offset: 1

Gregory A. Rosenthal, Mar 10 2013

For the actual numbers, see A031877 and their reversals in A008919. See especially the comments in A008919.

			The smallest examples of such numbers are 8712 and 9801 (so a(n)=0 for n < 4, a(4) = 2); 87912 and 98901 (so a(5) = 2); and 879912 and 989901 (so a(6) = 2).

W. W. R. Ball and H. S. M. Coxeter. Mathematical Recreations and Essays, Macmillan, New York, 1939, page 13; Dover, New York, 13th ed. 1987, pp. 14-15.
H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions D'Organisation, Paris, 1984.
Heinrich Dörrie, Mathematische Miniaturen, Ferdinand Hirt, Breslau, Germany, 1943; see pages 337-339.
M. Gardner, Mathematical Magic Show, Vintage Books, 1978, pp. 203, 204, 211, 212.
C. A. Grimm and D. W. Ballew, Reversible multiples, J. Rec. Math. 8 (1975-1976), 89-91.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, London, 1986, Entry 1089.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Yuhong Guo, Some Identities for Palindromic Compositions Without 2's, Journal of Mathematical Research with Applications 38.2 (2018): 130-136.
G. H. Hardy, A Mathematician's Apology, Cambridge Univ. Press, 1940, reprinted 2000, pp. 24-25.
J. Jonesco (proposer), E.-N. Barisien and [no initials given] Welsch (solvers), Problem 1622, L'Intermédiaire des mathématiciens, VI (1899), p. 200; L'Intermédiaire des mathématiciens, XV (1908), pp. 132-133, pp. 278-279 (in French).
T. J. Kaczynski, Note on a Problem of Alan Sutcliffe, Math. Mag., 41 (1968), 84-86.
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. Also arXiv:math/0511366 [math.HO], 2005-2006.
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Alan Sutcliffe, Integers That Are Multiplied When Their Digits Are Reversed, Mathematics Magazine, 39 (1966), 282-287.
Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.
Anne Ludington Young, Trees for k-reverse multiples, Fib. Q., 30 (1992), 166-174.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A001232, A008918, A008919, A031877, A103609, A222809, A222810, A222811, A222812.

See also A222817, A222818, A222819, A222820.

[0] cat [2*Fibonacci(Floor((n-2)/2)): n in [2..60]]; // Vincenzo Librandi, Jun 18 2013

Join[{0}, Table[2 Fibonacci[Floor[(n-2)/2]], {n, 2, 60}]] (* Vincenzo Librandi, Jun 18 2013 *)

def A214927(n): return 2*(fibonacci((n-2)//2) -int(n==1))
[A214927(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

a(n) = 2*Fibonacci(floor((n-2)/2)) = 2*A103609(n-2), for n > 1.

G.f.: 2*x^4*(1+x) / (1-x^2-x^4). - Colin Barker, Dec 31 2013

Formula, more terms and additional references and links from N. J. A. Sloane, Mar 11 2013

A001232 Numbers k such that 9*k = (k written backwards), k > 0.

Original entry on oeis.org

1089, 10989, 109989, 1099989, 10891089, 10999989, 108901089, 109999989, 1089001089, 1098910989, 1099999989, 10890001089, 10989010989, 10999999989, 108900001089, 108910891089, 109890010989, 109989109989, 109999999989, 1089000001089, 1089109891089

Offset: 1

N. J. A. Sloane and Simon Plouffe

This sequence contains the least n-digit non-palindromic number which is a factor of its reversal. Quotient is always 9. - Lekraj Beedassy, Jun 11 2004. (But it contains many other numbers as well. - N. J. A. Sloane, Jul 02 2013)
Nonzero fixed points of the map which sends x to x - reverse(x) if that is nonnegative, otherwise to x + reverse(x). - Sébastien Dumortier, Nov 05 2006. (Clarified comment, see A124074. - Ray Chandler, Oct 11 2017)
Numbers k such that reversal(k)=reversal(k+reversal(k)). Also numbers k such that reversal(k)=reversal(10*k-reversal(k)). - Farideh Firoozbakht, Jun 11 2010
From M. F. Hasler, Oct 04 2022: (Start)
(1) The first digit of any term must be 1, otherwise multiplication by 9 yields one more digit. For the same reason, no "overflow" must occur from the second to the first digit, so the last digit must be 9.
(2) Continuing the reasoning "from right to left" implies that the trailing nonzero digits must be ...9*89, where 9* means any nonnegative number of consecutive digits 9, preceded by a digit 0, which must be preceded by a digit 1. This implies that the initial and also final digits of any term must be 109*89. We might call a term of this form a "primitive" term. So there is exactly one primitive term b(k) = 11*10^(k-2)-11 with k digits, for all k >= 4.
(3) All terms of the sequence are a "symmetric" concatenation of such b(k)'s, "spaced out" with any number of digits 0, also in a symmetrical way: For any n >= 1, let k = (k[1], ..., k[n]) with k[n+1-j] = k[j] >= 4, and m = (m[1], ..., m[n-1]) (possibly of length 0) with m[n-j] = m[j] >= 0, then N = concat(b(k[j])*10^m[j], 1 <= j < n; k[n]) is a term of the sequence, and this yields all terms of the sequence. (For example, with 1089 we also have 1089{0...0}1089 and 1089,001089,001089, etc.) (End)

			1089*9 = 9801.

H. Camous, Jouer Avec Les Maths, "Cardinaux Réversibles", Section I, Problem 6, pp. 27, 37-38; Les Editions d'Organisation, Paris, 1984.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 41.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, under #1089.

Ray Chandler, Table of n, a(n) for n = 1..10000
C. A. Van Cott, The Integer Hokey Pokey, Math Horizons, Vol. 28, pp. 24-27, November 2020.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
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]
Simon Weisgerber, Value Judgments in Mathematics: GH Hardy and the (Non-)seriousness of Mathematical Theorems, Global Phil. (2024) Vol. 34, Art. No. 1. See p. 8.

Cf. A008918, A008919, A193434, A222814, A222815, A031877, A124074.

Cf. A002275, A002283, A004086, A094707.

Rest@Select[FromDigits /@ Tuples[{0, 99}, 11], IntegerDigits[9*#] == Reverse@IntegerDigits[#] &] (* Arkadiusz Wesolowski, Aug 14 2012 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; 99#&/@Flatten[Table[ FromDigits/@ Select[Tuples[{0,1},n],okQ],{n,20}]] (* Harvey P. Dale, Jul 03 2013 *)

isok(n) = 9*n == eval(concat(Vecrev(Str(n)))); \\ Michel Marcus, Feb 21 2015

{A001232_row(n, L(v, s=0)=for(i=1, #v, s*=10^v[i]; i%2 && s+=10^v[i]\900); s)=if(n<4, [], L, Set(apply(L, self()(n, 0)))*99, L=List([[n]]); for(k=4, n\2, listput(L,[k,n-2*k,k]); for(p=0, n\2-k, foreach(self()(n-(k+p)*2, 0), M, listput(L, concat([[k, p], M, [p, k]]))))); L)} \\ List of n-digit terms. - M. F. Hasler, Oct 04 2022
concat(apply(A001232_row, [1..14]))

def A001232_row(n, r=11): # list of n-digit terms
    L = [] if n<4 else [[n]]
    for L1 in range(4, n//2+1):
        L.append([L1, n-2*L1, L1])
        L.extend([L1,L2]+M+[L2,L1] for L2 in range(n//2-1-L1)
                                     for M in A001232_row(n-(L1+L2)*2, 0))
    if not r: return L
    def f(L, s=0):
        for k,L in enumerate(L):
            s *= 10**L
            if not k%2: s += 10**(L-2)-1
        return r*s
    return sorted(map(f, A001232_row(n, 0))) # M. F. Hasler, Oct 04 2022

Theorem: Terms in this sequence have the form 99*m, where the decimal representation of m contains only 1's and 0's, is palindromic and contains no singleton 1's or 0's. Hence contains Fib(floor(k/2)-1) k-digit terms, k >= 4. - David W. Wilson, Dec 15 1997
a(A094707(n)) = 11*(10^n - 1) = 11*A002283(n) = 99*A002275(n), for n>1. - Lekraj Beedassy, Jun 11 2004. (Restored from history and corrected. - Ray Chandler, Oct 11 2017)
a(n) = 99*A061851(n) = A008918(n)/2. - M. F. Hasler, Oct 06 2022

Corrected and extended by David W. Wilson, Aug 15 1996, Dec 15 1997
a(20)-a(21) from Arkadiusz Wesolowski, Aug 14 2012
a(1..10^4) in b-file double-checked with independent code by M. F. Hasler, Oct 04 2022

A008918 Numbers k such that 4*k = (k written backwards), k > 0.

Original entry on oeis.org

2178, 21978, 219978, 2199978, 21782178, 21999978, 217802178, 219999978, 2178002178, 2197821978, 2199999978, 21780002178, 21978021978, 21999999978, 217800002178, 217821782178, 219780021978, 219978219978, 219999999978, 2178000002178, 2178219782178

Offset: 1

N. J. A. Sloane

There are Fibonacci(floor((k-2)/2)) terms with k digits (this is essentially A103609). - Ray Chandler, Oct 12 2017

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 41-42.
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 200 terms from Vincenzo Librandi)
C. A. Van Cott, The Integer Hokey Pokey, Math Horizons, Vol. 28, pp. 24-27, November 2020.
L. H. Kendrick, Young Graphs: 1089 et al., J. Int. Seq. 18 (2015) 15.9.7.
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]

Cf. A001232, A193434, A008918, A008919, A222814, A222815, A031877.

Rest@Select[FromDigits /@ Tuples[{0, 198}, 11], IntegerDigits[4*#] == Reverse@IntegerDigits[#] &] (* Arkadiusz Wesolowski, Aug 14 2012 *)
okQ[t_]:=t==Reverse[t]&&First[t]!=0&&Min[Length/@Split[t]]>1; 198#&/@ Flatten[ Table[FromDigits/@Select[Tuples[{0,1},n],okQ],{n,20}]] (* Harvey P. Dale, Jul 03 2013 *)

rev(n) = (eval(concat(Vecrev(Str(n)))));
isok(n) = rev(n) == 4*n; \\ Michel Marcus, Sep 13 2015

Theorem (David W. Wilson): a(n) = 2*A001232(n).

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

a(20)-a(21) from Arkadiusz Wesolowski, Aug 14 2012

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

Original entry on oeis.org

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

A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

16, 96, 416, 496, 576, 2016, 2496, 2976, 10016, 10416, 12096, 12496, 14976, 50016, 52416, 60096, 62496, 74976, 250016, 252016, 260416, 262416, 300096, 302096, 310496, 312496, 360576, 374976, 1250016, 1262016, 1300416, 1312416, 1500096, 1512096, 1550496

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

n/m must be either 2 or 4. - Robert Israel, Apr 22 2021

			a(1) = 16 because: 16 in base 5 is 31; 31 reversed is 13; 13 converted back to base 10 is 8 and 16 mod 8 = 0.

Robert Israel, Table of n, a(n) for n = 1..10000
C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link?]

Cf. A091077 (same in base 3), A091078 (base 4), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

Original entry on oeis.org

105, 108, 405, 510, 540, 702, 703, 810, 1001, 1005, 1008, 1020, 1050, 1053, 1080, 2002, 2016, 2025, 2040, 2050, 2079, 2100, 2106, 3003, 3024, 3042, 3045, 3060, 3105, 3402, 3510, 4004, 4005, 4050, 4070, 4080, 4200, 5005, 5010, 5040

Offset: 1

Chuck Seggelin, Nov 21 2003

Trivial permutations are identified as (1) permutation = n, or (2) when n mod 10=0, permutations of n's digits which result in shifting only trailing zeros to the most significant side of n where they drop off, such that permutation = n/10^z, where z <= the number of trailing zeros of n. So if n were 1809000, the following permutations would be excluded as trivial: 1809000, 0180900, 0018090, 0001809.

A031877 (numbers which are multiples of their reversals) and both A084687 and A090053 (numbers divided by number formed by sorting their digits), are subsets of this sequence. This sequence differentiates itself by including terms such as 7425 which is divided by 2475 (a rearrangement of 7425's digits that is neither a reversal or an ascending sort.)

			a(27)=3045 because 3045 is divisible by 435, a nontrivial permutation of 3045. (0435)

C. Seggelin, Numbers Divisible by Digit Permutations.

Cf. A031877, A084687, A090056, A090057, A090058, A090059, A090060.

A091077 Numbers n which when converted to base 3, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

64, 208, 640, 1936, 5248, 5824, 15616, 17488, 46720, 50752, 52480, 140032, 151840, 157456, 419968, 425152, 455104, 467200, 472384, 1259776, 1276624, 1364896, 1400320, 1417168, 3779200, 3794752, 3831040, 4094272, 4109824, 4199680, 4235968, 4251520

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 64 because: 64 in base 3 is 2101; 2101 reversed is 1012; 1012 converted back to base 10 is 32 and 64 mod 32 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091078 (same in base 4), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

isok(n, b=3) = {m = subst(Polrev(digits(n, b)), x, b); if (n % m, return(0));if ((n/m == 1), return (0)); vq = valuation(n, b); if (! vq, return (1)); qq = subst(Polrev(digits(m,b)), x, b); if (n == b^vq*qq, return (0)); return (1);} \\ Michel Marcus, Oct 10 2014

More terms from Michel Marcus, Oct 10 2014

A091078 Numbers n which when converted to base 4, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

225, 945, 3825, 15345, 57825, 61425, 230625, 245745, 921825, 968625, 983025, 368, 6625, 3871665, 3932145, 14745825, 14803425, 15483825, 15671025, 15728625, 589826, 25, 59224545, 61932465, 62672625, 62914545

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 225 because: 225 in base 4 is 3201; 3201 reversed is 1023; 1023 converted back to base 10 is 75 and 225 mod 75 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091077 (same in base 3), A091079 (base 5), A091080 (base 6), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

/* See A091077 and use PARI script with b=4 */

More terms from Michel Marcus, Oct 10 2014

A091080 Numbers n which when converted to base 6, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

980, 1225, 6020, 7525, 36260, 45325, 217700, 272125, 1271060, 1306340, 1588825, 1632925, 7621460, 7838180, 9526825, 9797725, 45723860, 46817540, 47029220, 57154825, 58521925, 58786525

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 980 because: 980 in base 6 is 4312; 4312 reversed is 2134; 2134 converted back to base 10 is 490 and 980 mod 490 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091081 (base 7), A091082 (base 8), A091083 (base 9), A031877 (base 10).

/* See A091077 and use PARI script with b=6 */

More terms from Michel Marcus, Oct 10 2014

A091081 Numbers n which when converted to base 7, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

Original entry on oeis.org

36, 288, 1800, 2052, 2304, 12384, 14400, 16416, 86472, 88236, 99072, 100836, 115200, 605088, 619200, 691776, 705888, 806688, 4235400, 4247748, 4323600, 4335948, 4840704, 4853052, 4928904, 4941252, 5534208, 5647104, 29647584, 29746368, 30254400, 30353184

Offset: 1

Chuck Seggelin, Dec 18 2003

Trivial cases are those numbers which upon conversion result in a number which is palindromic (m = reverse(m)), or a palindrome plus trailing zeros such that m = reverse(m)*10^z where z=number of lost zeros. Nontrivial digit loss occurs when a converted number has trailing zeros that drop off when the number is reversed.

			a(1) = 36 because: 36 in base 7 is 51; 51 reversed is 15; 15 converted back to base 10 is 12 and 36 mod 12 = 0.

C. Seggelin, Numbers Divisible by Digit Permutations. [Broken link]

Cf. A091077 (same in base 3), A091078 (base 4), A091079 (base 5), A091080 (base 6), A091082 (base 8), A091083 (base 9), A031877 (base 10).

/* See A091077 and use PARI script with b=7 */

More terms from Michel Marcus, Oct 10 2014

cp's OEIS Frontend

A214927 Number of n-digit numbers N that do not end with 0 and are such that the reversal of N divides N but is different from N.

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A001232 Numbers k such that 9*k = (k written backwards), k > 0.

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A008918 Numbers k such that 4*k = (k written backwards), k > 0.

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

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A091079 Numbers n which when converted to base 5, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.

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Maple

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A090055 Numbers n divisible by at least one nontrivial permutation (rearrangement) of the digits of n.

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A091077 Numbers n which when converted to base 3, reversed and converted back to base 10 yield a number m such that n mod m = 0. Cases which are trivial or result in digit loss are excluded.