A222810 -id:A222810 - 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

A222811 Number of n-digit numbers N such that the reversal of N divides N.

Original entry on oeis.org

9, 18, 111, 212, 1128, 2067, 11123, 20270, 110440, 200971, 1101475, 2003592, 11005388

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.

Even terms are roughly double the previous term. - David Consiglio, Jr., Mar 22 2013

			Some of the smallest solutions are:
[1, 2, 3, 4, 5, 6, 7, 8, 9] (so a(1) = 9),
[10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99] (so a(2) = 18),
[100, 101, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 200, 202, 212, 220, 222, 232, 242, 252, 262, 272, 282, 292, 300, 303, 313, 323, 330, 333, 343, 353, ...]

David Consiglio, Jr., C++ program
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]

Cf. A214927, A222809, A222810, A222812.

a(8) - a(11) from David Consiglio, Jr., Mar 22 2013

a(12)-a(13) from Giovanni Resta, Apr 01 2013

A222809 Number of n-digit numbers N such that the reversal of N divides N but is different from N.

Original entry on oeis.org

0, 9, 21, 122, 228, 1167, 2123, 11270, 20440, 110971, 201475, 1103592, 2005388

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.
Conjecture: a(n) = A222811(n) - 9*10^floor((n-1)/2). - Lars Blomberg, Jul 03 2014
Proof of this conjecture: A number N with n digits which equals its own reversal has the first and identical last digit from {1, 2, ..., 9}. If n is even the other n-2 digits come in n/2 - 1 pairs of equal numbers from {0, 1, ..., 9}. If n is odd with n >= 3 then the other n-2 numbers come in (n-3)/2 pairs of equal numbers from {0, 1, ..., 9} and an additional middle digit also from {0, 1, ..., 9}. Therefore there are 9*10^(n/2-1) such numbers N for even n, and 9*10^((n-1)/2) for odd n, fitting 9*10^floor((n-1)/2). - Wolfdieter Lang, Jul 13 2014

			Some of the smallest solutions are:
[10, 20, 30, 40, 50, 60, 70, 80, 90] (so a(2) = 9),
[100, 110, 200, 220, 300, 330, 400, 440, 500, 510, 540, 550, 600, 660, 700, 770, 800, 810, 880, 900, 990] (so a(3) = 21),
[1000, 1010, 1100, 1110, 1210, 1310, 1410, ...].

Cf. A214927, A222810, A222811, A222812.

a(n) = sum(i=10^(n-1), 10^n-1, (irev=eval(concat(Vecrev(Str(i))))) && irev!=i && !(i % irev)); \\ Michel Marcus, Jul 03 2014

a(7)-a(12) from Lars Blomberg, Jul 03 2014

a(13) from Giovanni Resta, Aug 15 2019

A222812 Number of n-digit numbers N such that the number formed by some nontrivial permutation of the digits of N divides N.

Original entry on oeis.org

0, 18, 329, 5000, 65931, 779504, 8517616, 88555255, 897147508, 8997325290, 90000000000, 900000000000, 9000000000000, 90000000000000, 900000000000000, 9000000000000000, 90000000000000000, 900000000000000000, 9000000000000000000, 90000000000000000000

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.

			Some of the smallest solutions are:
[10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99] (so a(2) = 18),
[100, 101, 105, 108, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 130, 131, 133, 140, 141, 144, 150, 151, 155, 160, 161, 166, 170, 171, 177, 180, 181, 188, 190, 191, 199, 200, 202, 210, 211, 212, 220, 221, 222, 223, 224, ...]
Note that 11 is in the sequence because permuting the two digits gives 11, and 11 divides 11.

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A214927, A222809, A222810, A222811.

a(n) = 9 * 10^(n-1) for n >= 11. - Hiroaki Yamanouchi, Sep 03 2014.

G.f.: x^2*(26747100*x^9 +25850210*x^8 +11594958*x^7 +3379095*x^6 +722576*x^5 +120194*x^4 +15931*x^3 +1710*x^2 +149*x +18)/(1-10*x). - Robert Israel, Sep 03 2014

a(7)-a(20) from Hiroaki Yamanouchi, Sep 03 2014

A223080 Numbers n with distinct digits such that the reversal of n divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 510, 540, 810, 5610, 5940, 8712, 8910, 9801, 65340, 87120, 87912, 659340, 879120

Offset: 1

N. J. A. Sloane, Mar 20 2013

Suggested by A214927.
There are no terms with 7 digits.
Sequence is complete. - Giovanni Resta, Mar 20 2013

			8712 reversed is 2178, which divides 8712.

Cf. A222810, A214927, A223081, A223082.

Select[Range[10^6],Max[DigitCount[#]]==1&&Divisible[#,IntegerReverse[#]]&] (* Harvey P. Dale, Jun 21 2022 *)

A223081 Numbers n with distinct digits such that n divides the reversal of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1089, 2178, 21978

Offset: 1

N. J. A. Sloane, Mar 20 2013

The definition implies that n does not end in 0.
There are no terms with 6 digits.
Sequence is complete. - Giovanni Resta, Mar 20 2013

Cf. A214927, A223080, A223082, A222810.

Select[FromDigits/@(Flatten[Permutations/@Subsets[Range[0,9],5],1]), Divisible[ IntegerReverse[#],#]&]//Quiet//Union (* Harvey P. Dale, Jan 12 2020 *)

A223082 Number of n-digit numbers N with distinct digits such that N divides the reversal of N.

Original entry on oeis.org

9, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 20 2013

The definition implies that N does not end in 0.
The numbers N are listed in A223081.
a(n) = 0 for n > 5.

Cf. A223081 (the actual numbers), A214927, A222810, A223080.

Entry revised by N. J. A. Sloane, Jul 04 2021

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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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A222811 Number of n-digit numbers N such that the reversal of N divides N.

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A222809 Number of n-digit numbers N such that the reversal of N divides N but is different from N.

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A222812 Number of n-digit numbers N such that the number formed by some nontrivial permutation of the digits of N divides N.

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A223080 Numbers n with distinct digits such that the reversal of n divides n.

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A223081 Numbers n with distinct digits such that n divides the reversal of n.

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A223082 Number of n-digit numbers N with distinct digits such that N divides the reversal of N.

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