A222809 -id:A222809 - 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

A222810 Number of n-digit numbers N with distinct digits such that the reversal of N divides N.

Original entry on oeis.org

9, 9, 3, 5, 3, 2, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 10 2013

Suggested by A214927.

a(n)=0 for all n > 6.

			Solutions with 1 through 6 digits:
[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],

Cf. A214927, A222809, A222811, A222812.

For the actual numbers see A223080.

import collections
col = []
count = 0
for n in range(0, 9):
    a = 10**n
    stop = 10**(n+1)
    while a < stop:
        b = str(a)
        c = list(b)
        d = c[::-1]
        e = int("".join(c))
        f = int("".join(d))
        counter = collections.Counter(c)
        if e % f == 0 and counter.most_common(1)[0][1] == 1:
            count += 1
            col.append(a)
        a += 1
    print(n+1, " digits: ", count, " elements: ", col)
    count = 0
    col = []
# David Consiglio, Jr., Dec 04 2014

a(8)-a(9) from David Consiglio, Jr., Dec 04 2014

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

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

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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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A222810 Number of n-digit numbers N with distinct digits such that the reversal of N divides N.

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