A008919 -id:A008919 - 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

A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

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

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.").

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.
N. J. A. Sloane, 2178 And All That [Local copy]
Eric Weisstein's World of Mathematics, Reversal.

See A008919 for reversals (this is the main entry for the problem).
Cf. A169824, A214927, A103609.
Union of A222814 and A222815.
Subsequence of A118959.

a031877_list = [x | x <- [1..], x `mod` 10 > 0,
                    let x' = a004086 x, x' /= x && x `mod` x' == 0]
-- Reinhard Zumkeller, Jul 15 2013

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 *)

is_A031877(n)={n%10 && n%A004086(n)==0 && n>A004086(n)} \\ M. F. Hasler, Aug 18 2014

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

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

More terms from Jud McCranie, Aug 15 2001

More terms from Sam Mathers, Aug 18 2014

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

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

A245680 Numbers x whose digits can be permuted to produce a multiple of x.

Original entry on oeis.org

1035, 1089, 1359, 1386, 1782, 2178, 2475, 10035, 10089, 10350, 10449, 10890, 10899, 10989, 11688, 11883, 12375, 12903, 13029, 13359, 13449, 13590, 13599, 13659, 13860, 13986, 14085, 14247, 14724, 14859, 15192, 16782, 17604, 17802, 17820, 17832, 17982, 18027

Offset: 1

Paolo P. Lava, Jul 29 2014

A008919 is a subset of this sequence.
Every element of the sequence is divisible by 3. - Emmanuel Vantieghem, Oct 27 2015
It is an obvious fact that if a(n) is the n-th term of the sequence, then a(n)*(10^k) is also a member of the sequence for all k > 0. - Altug Alkan, Nov 01 2015

			A permutation of 1782 is 7128 and 7128 / 1782 = 4.
A permutation of 11688 is 81816 and 81816 / 11688 = 7.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A008919, A096092, A096093, A245682.

P:=proc(q) local a, b, c, i, j, k, n, t; for n from 1 to q do a:=n; b:=[];
while a>0 do b:=[a mod 10, op(b)]; a:=trunc(a/10); od;
t:=0; for i from 2 to 9 do a:=i*n; c:=[];
while a>0 do c:=[a mod 10, op(c)]; a:=trunc(a/10); od;
if sort(b)=sort(c) then print(n); break; fi; od; od; end: P(10^6);
# Alternative:
N:= 100: # to get the first N entries
count:= 0:
for x from 10 while count < N do
  M:= 10^(ilog10(x)+1)-1;
  L:= sort(convert(x,base,10));
  for i from 2 to floor(M/x) do
    Lp:= sort(convert(i*x,base,10));
    if Lp = L then
      count:= count+1;
      A[count]:= x;
      break;
    fi
   od
od:
seq(A[i],i=1..count); # Robert Israel, Jul 29 2014

fQ[n_] := AnyTrue[Rest[FromDigits /@ Permutations[IntegerDigits@ n]], Divisible[#, n] &]; Select[Range@ 20000, fQ] (* Michael De Vlieger, Oct 27 2015, Version 10 *)

for(n=1,10^8,d=vecsort(digits(n));p=0;for(k=2,9,dd=vecsort(digits(n*k));if(d==dd,p++;break));if(p>0,print1(n,", "))) \\ quicker program Derek Orr, Jul 29 2014

import itertools
from itertools import permutations
for n in range(1,10**5):
  plist = list(permutations(str(n)))
  for i in plist:
    num = ''
    for j in range(len(i)):
      num += i[j]
    if int(num)%n==0 and int(num)/n > 1:
      print(n,end=', ') # Derek Orr, Jul 29 2014

A245682 Numbers x whose digits can be permuted to produce more than a single multiple of x.

Original entry on oeis.org

123876, 142857, 153846, 230769, 285714, 1028574, 1218753, 1238760, 1239876, 1246878, 1294857, 1402857, 1420785, 1425897, 1428507, 1428570, 1428597, 1428705, 1429857, 1485792, 1492857, 1538460, 1539846, 1570284, 1584297, 2300769, 2307690, 2307699, 2309769, 2857014, 2857140, 2859714, 2985714, 10028574, 10178649

Offset: 1

Paolo P. Lava, Jul 29 2014

It is a subset of A245680.

If x < 10^d is in the sequence, then so are x*10^j*(1+10^d+...+10^k*d) for all nonnegative integers j and k. - Robert Israel, Jul 29 2014

			Two permutations of 123876 are 371628, 867132  and  371628 / 123876 = 3, 867132  / 123876 = 7.
Five permutations of 142857 are 285714, 428571, 571428, 714285, 857142 and 285714 / 142857 = 2, 428571 / 142857 = 3, 571428 / 142857 = 4, 714285 / 142857 = 5, 857142 / 142857 = 6.

Paolo P. Lava, Table of n, a(n) for n = 1..200

Cf. A008919, A096092, A096093, A245680.

P:=proc(q) local a,b,c,i,j,k,n,t; for n from 1 to q do a:=n; b:=[];
while a>0 do b:=[a mod 10,op(b)]; a:=trunc(a/10); od;
t:=0; for i from 2 to 9 do a:=i*n; c:=[];
while a>0 do c:=[a mod 10,op(c)]; a:=trunc(a/10); od;
if sort(b)=sort(c) then t:=t+1; fi; if t>1 then print(n); break;
fi; od; od; end: P(10^10);
# Alternative
N:= 10: # get a(1) to a(N)
count:= 0:
for x from 10 while count < N do
  M:= 10^(ilog10(x)+1)-1;
  L:= sort(convert(x,base,10));
  mults:= 0;
  for i from 2 to floor(M/x) do
    Lp:= sort(convert(i*x,base,10));
    if Lp = L then
      mults:= mults+1;
      if mults = 2 then
        count:= count+1;
        A[count]:= x;
        print(x);
        break;
      fi
    fi
   od
od:
seq(A[i],i=1..count); # Robert Israel, Jul 29 2014

for(n=1,10^8,d=vecsort(digits(n));p=0;for(k=2,9,dd=vecsort(digits(n*k));if(d==dd,p++));if(p>1,print1(n,", "))) \\ faster program Derek Orr, Jul 29 2014

import itertools
from itertools import permutations
for n in range(1,10**8):
  plist = list(permutations(str(n)))
  count = 0
  lst = []
  for i in plist:
    num = ''
    for j in range(len(i)):
      num += i[j]
    if int(num)%n==0 and int(num)/n > 1:
      if int(num) not in lst:
        lst.append(int(num))
        count += 1
  if count > 1:
    print(n,end=', ') # Derek Orr, Jul 29 2014

a(7) to a(10) from Robert Israel, Jul 29 2014

a(11) - a(35) from Derek Orr, Jul 29 2014

A061467 Remainder when the larger of n and its reverse is divided by the smaller.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 5, 13, 6, 13, 3, 9, 15, 0, 9, 0, 9, 18, 2, 10, 18, 26, 5, 0, 5, 9, 0, 9, 18, 27, 36, 7, 15, 0, 13, 18, 9, 0, 9, 18, 27, 36, 45, 0, 6, 2, 18, 9, 0, 9, 18, 27, 36, 0, 13, 10, 27, 18, 9, 0, 9, 18, 27, 0, 3, 18, 36, 27, 18, 9, 0, 9, 18, 0, 9, 26, 7, 36, 27

Offset: 0

Erich Friedman, Jun 16 2001

a(n)=0 if n is in A002113, A008919 or A118959. - Robert Israel, Jul 18 2019

			a(12)=9 since 21/12 = 1 with remainder 9.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A002113, A004086, A008919, A118959.

a061467 0 = 0
a061467 n = mod (max n n') (min n n') where n' = a004086 n
-- Reinhard Zumkeller, Dec 31 2013

l := {} For[i = 1, i < 100, i++, x := FromDigits[Reverse[IntegerDigits[i]]]; If[x >= i, AppendTo[l, Mod[x, i]], AppendTo[l, Mod[i, x]]]] l (* Jake Foster, Jun 05 2008 *)
rln[n_]:=Module[{r=IntegerReverse[n]},If[r>n,Mod[r,n],Mod[n,r]]]; Join[ {0}, Array[rln,90]] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Apr 03 2016 *)

{ for (n=0, 1000, x=n; r=0; while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); p=max(n, r); q=min(n, r); write("b061467.txt", n, " ", p%q) ) } \\ Harry J. Smith, Jul 23 2009

A061852 Digital representation of m contains only either 1's or 2's (but not both 1's and 2's) and 0's, is palindromic and contains no singleton 2's, 1's or 0's.

Original entry on oeis.org

11, 22, 111, 222, 1111, 2222, 11111, 22222, 110011, 111111, 220022, 222222, 1100011, 1111111, 2200022, 2222222, 11000011, 11100111, 11111111, 22000022, 22200222, 22222222, 110000011, 111000111, 111111111, 220000022, 222000222

Offset: 1

Henry Bottomley, May 10 2001

			From _M. F. Hasler_, Oct 17 2022: (Start)
Written in rows, where each row has terms of given length and given digit set (either no 2 or no 1), the sequence starts:
  row | terms
------+------------------------------------
    1 | 11
    2 | 22
    3 | 111
    4 | 222
    5 | 1111
    6 | 2222
    7 | 11111
    8 | 22222
    9 | 110011, 111111
   10 | 220022, 222222
Then for any n >= 1, row 2n = 2*(row 2n-1) and row 2n-1 = (terms in A061851 with n+1 digits), and the number of terms in row n is Fibonacci(ceiling(n/4)) = A000045(A002265(n+3)), and their length (number of digits) is ceiling(n/2)+1 = floor((n+3)/2). (End)

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A008919.
Union of A061851 and twice A061851.
Number of terms with k digits is 2*Fibonacci(floor(k/2)) = 2*A000045(A004526(k)) = A006355(floor(k/2)+1).

A061852_row(n)=A061851_row(n\/2+1)*(2-n%2) \\ Note: This refers to rows as defined in EXAMPLE, while A061851_row gives the n-digit terms. - M. F. Hasler, Oct 17 2022

a(n) = A008919(n)/99.

A071685 Non-palindromic numbers n, not divisible by 10, such that either n divides R(n) or R(n) divides n, where R(n) is the digit-reversal of n.

Original entry on oeis.org

1089, 2178, 8712, 9801, 10989, 21978, 87912, 98901, 109989, 219978, 879912, 989901, 1099989, 2199978, 8799912, 9899901, 10891089, 10999989, 21782178, 21999978, 87128712, 87999912, 98019801, 98999901, 108901089, 109999989

Offset: 1

Labos Elemer, Jun 03 2002

The quotient R(n)/n or n/R(n) is always 4 or 9.
This is the union of the four sequence A001232, A222814, A008918, A222815. Equivalently, the union of A008919 and A031877.
There are 4*Fibonacci(floor((n-2)/2)) terms with n digits (this is 2*A214927 or essentially 4*A103609). - Ray Chandler, Oct 12 2017
Conjecture: every term mod 100 is equal to 1, 12, 78, or 89. - Harvey P. Dale, Dec 13 2017

			Palindromic solutions like 12021 or also solutions divisible by 10 were filtered out like {8380,838; q=10} or {8400,48; q=175}. In case of m>R(m), q=m/R(m)=4 or 9.

Ray Chandler, Table of n, a(n) for n = 1..10000
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, A222814, A008918, A222815, A008919, A031877.

Cf. A004086, A055483, A069554, A103609, A214927.

nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] ed[x_] := IntegerDigits[x] red[x_] := Reverse[IntegerDigits[x]] Do[s=Mod[Max[{n, tn[red[n]]}], Min[{n, r=tn[red[n]]}]]; If[Equal[s, 0]&&!Equal[Mod[n, 10], 0] &&!Equal[n, r], Print[{n, r/n}]], {n, 1, 1000000}]
npnQ[n_]:=Module[{r=IntegerReverse[n]},!PalindromeQ[n]&&!Divisible[ n,10] &&(Mod[n,r]==0||Mod[r,n]==0)]; Select[Range[11*10^7],npnQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2017 *)

x = q*R(x), q is an integer q<>1, q<>10^j and neither of x or R(x) is divisible by 10.

Corrected and extended by Harvey P. Dale, Jul 01 2013
Edited by N. J. A. Sloane, Jul 02 2013
Missing terms inserted by Ray Chandler, Oct 09 2017
Incorrect comment removed by Ray Chandler, Oct 12 2017

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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A031877 Nontrivial reversal numbers (numbers which are integer multiples of their reversals), excluding palindromic numbers and multiples of 10.

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

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

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A245680 Numbers x whose digits can be permuted to produce a multiple of x.

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