A222819 -id:A222819 - 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

A222817 Irregular triangle read by rows: row n gives list of nontrivial reverse multipliers for base n.

Original entry on oeis.org

2, 3, 2, 4, 2, 5, 3, 6, 2, 3, 5, 7, 2, 4, 8, 4, 9, 2, 3, 5, 7, 10, 2, 3, 5, 11, 5, 6, 12, 2, 3, 4, 6, 9, 13, 2, 3, 4, 7, 11, 14, 3, 7, 15, 2, 4, 5, 8, 10, 11, 16, 2, 5, 7, 8, 17, 3, 4, 6, 7, 9, 14, 18, 2, 3, 4, 6, 9, 13, 19

Offset: 3

N. J. A. Sloane, Mar 13 2013

If there is a number m such that the reversal of m in base n is c times m, then c is called a reverse multiplier for n. For example, 2 is a reverse multiplier for base n=5, since 8 (base 10) = 13 (base 5), and 2*8 = 16 (base 10) = 31 (base 5).
The trivial reverse multiplier 1 is excluded.
The last entry in each row is n-1; the number of terms in row n is A222819(n).

			Triangle begins:
  2,
  3,
  2,4,
  2,5,
  3,6,
  2,3,5,7,
  2,4,8,
  4,9,
  2,3,5,7,10,
  2,3,5,11,
  5,6,12,
  2,3,4,6,9,13,
  2,3,4,7,11,14,
  3,7,15,
  ...

N. J. A. Sloane, Table giving n, list of nontrivial reverse multipliers, for n = 3..100
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]
Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.

Cf. A214927, A222818, A222819, A222820.

See A214927 for other cross-references.

A222820 a(n) is the number of reverse multipliers for base n.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 4, 3, 6, 5, 4, 7, 7, 4, 8, 6, 8, 8, 7, 6, 11, 11, 6, 8, 9, 6, 13, 12, 10, 13, 6, 9, 14, 10, 9, 13, 17, 9, 15, 12, 13, 17, 13, 11, 20, 16, 12, 12

Offset: 2

N. J. A. Sloane, Mar 13 2013

If there is a number m such that the reversal of m in base n is c times m, then c is called a reverse multiplier for n. For example, 2 is a reverse multiplier for base n=5, since 8 (base 10) = 13 (base 5), and 2*8 = 16 (base 10) = 31 (base 5).
The trivial reverse multiplier 1 is included.
a(n)-1 is the length of row n of A222817. - Michel Marcus, Apr 12 2020

For a complete list of references and links related to this problem see A214927.

N. J. A. Sloane, 2178 And All That, arXiv:1307.0453 [math.NT], 2013; see also, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.

Cf. A214927, A222817, A222818, A222819.

See A214927 for other cross-references.

A222818 Irregular triangle read by rows: row n gives list of reverse multipliers for base n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 2, 5, 1, 3, 6, 1, 2, 3, 5, 7, 1, 2, 4, 8, 1, 4, 9, 1, 2, 3, 5, 7, 10, 1, 2, 3, 5, 11, 1, 5, 6, 12, 1, 2, 3, 4, 6, 9, 13, 1, 2, 3, 4, 7, 11, 14, 1, 3, 7, 15, 1, 2, 4, 5, 8, 10, 11, 16, 1, 2, 5, 7, 8, 17, 1, 3, 4, 6, 7, 9, 14, 18, 1, 2, 3, 4, 6, 9, 13, 19

Offset: 2

N. J. A. Sloane, Mar 13 2013

If there is a number m such that the reversal of m in base n is c times m, then c is called a reverse multiplier for n. For example, 2 is a reverse multiplier for base n=5, since 8 (base 10) = 13 (base 5), and 2*8 = 16 (base 10) = 31 (base 5).
The trivial reverse multiplier 1 is included.
The last entry in each row is n-1; the number of terms in row n is A222820(n).

			Triangle begins:
  1,
  1,2,
  1,3,
  1,2,4,
  1,2,5,
  1,3,6,
  1,2,3,5,7,
  1,2,4,8,
  1,4,9,
  1,2,3,5,7,10,
  1,2,3,5,11,
  1,5,6,12,
  1,2,3,4,6,9,13,
  1,2,3,4,7,11,14,
  1,3,7,15
 ...

For a complete list of references and links related to this problem see A214927.

N. J. A. Sloane, Table giving n, number of nontrivial reverse multipliers, list of nontrivial reverse multipliers, for n = 3..50
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]
Anne Ludington Young, k-Reverse multiples, Fib. Q., 30 (1992), 126-132.

Cf. A214927, A222817, A222819, A222820.

See A214927 for other cross-references.

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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A222817 Irregular triangle read by rows: row n gives list of nontrivial reverse multipliers for base n.

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A222820 a(n) is the number of reverse multipliers for base n.

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A222818 Irregular triangle read by rows: row n gives list of reverse multipliers for base n.

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