A214927 -id:A214927 - cp's OEIS frontend

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.

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

A222816 Numbers k that divide the base-5 reversal of k.

Original entry on oeis.org

8, 48, 144, 208, 248, 744, 1008, 1248, 3744, 5008, 5208, 6048, 6248, 18744, 25008, 26208, 30048, 31248, 90144, 93744, 125008, 126008, 130208, 131208, 150048, 151048, 155248, 156248, 450144, 468744, 625008, 631008, 650208, 656208, 750048, 756048, 775248, 781248

Offset: 1

N. J. A. Sloane, Mar 13 2013

The base-B reversal of k is obtained by writing k in base B and reversing the order of the digits.

			8 in base 5 is 13_5; its reverse is 31_5, which is 16 in base 10, and 8 divides 16, so 8 is in the sequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..57
Leonard F. Klosinski and Dennis C. Smolarski, On the Reversing of Digits, Math. Mag., 42 (1969), 208-210. [Gives a(1)]
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. A214927, A091079.

Select[Range[10^6],Divisible[s=FromDigits[Reverse@IntegerDigits[#,5],5],#]&&s!=#&] (* Giorgos Kalogeropoulos, Aug 20 2021 *)

isok(n) = (rdn = subst(Polrev(digits(n,  5)), x, 5)) && (rdn != n) && (rdn % n) == 0; \\ Michel Marcus, Oct 10 2014

More terms from Alois P. Heinz, Mar 13 2013

A223078 Positive integers with the property that if the base-4 representation is reversed the result is three times the original number.

Original entry on oeis.org

75, 315, 1275, 5115, 19275, 20475, 76875, 81915, 307275, 322875, 327675, 1228875, 1290555, 1310715, 4915275, 4934475, 5161275, 5223675, 5242875, 19660875, 19741515, 20644155, 20890875, 20971515, 78643275, 78720075, 78969675, 82575675, 82652475, 83559675

Offset: 1

N. J. A. Sloane, Mar 14 2013

From Robert Israel, Apr 23 2019: (Start)
All terms are divisible by 15.
If x is a term and x < 4^k, then x*(4^k+1) is a term.  In particular the sequence is infinite. (End)

Robert Israel, Table of n, a(n) for n = 1..985

Cf. A173951, A223077, A223079, A214927.

rev4:= proc(n) local L,i;
  L:= convert(n,base,4);
  add(L[-i]*4^(i-1),i=1..nops(L))
end proc:
Res:= NULL:
for d from 2 to 15 do
  d1:= ceil(d/2); d2:= d-d1;
  for a from 4^(d1-1) to 4^d1/3 do
     b:= rev4(a)/3 mod 4^d2;
     x:= 4^d2*a+b;
     if rev4(x) = 3*x then Res:= Res, x; fi
od od:
Res; # Robert Israel, Apr 23 2019

Select[Range[84*10^6],3#==FromDigits[Reverse[IntegerDigits[#,4]],4]&] (* Harvey P. Dale, Mar 03 2018 *)

More terms from Alois P. Heinz, Mar 14 2013

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

A223089 Numbers n, written in base 8, with the property that if the base-8 representation of n is read backwards, the result is 5*n.

Original entry on oeis.org

1015, 11165, 102515, 112665, 1016015, 1127665, 10151015, 10252515, 11176165, 11277665, 101501015, 102617515, 111661165, 112777665, 1015001015, 1016016015, 1025252515, 1026267515, 1116511165, 1117526165, 1126762665, 1127777665

Offset: 1

N. J. A. Sloane, Mar 23 2013

For the decimal representations of these numbers, see A217742.

Theorem: There are 2^{floor(d/2)-2} terms with d base-8 digits.

			1015 reversed is 5101 = 5*1015 (calculating in base 8).

Cf. A214927, A217742.

A223079 Positive integers, written in base 4, with the property that if the base-4 representation is reversed the result is three times the original number.

Original entry on oeis.org

1023, 10323, 103323, 1033323, 10231023, 10333323, 102301023, 103333323, 1023001023, 1032310323, 1033333323, 10230001023, 10323010323, 10333333323, 102300001023, 102310231023, 103230010323, 103323103323, 103333333323

Offset: 1

N. J. A. Sloane, Mar 14 2013

For the decimal representations of these numbers see A223078.

Cf. A173951, A214927, A223078.

A224220 a(n) = smallest number k with property that if the base-n expansion of k is reversed, the result is a nontrivial multiple of k.

Original entry on oeis.org

32, 75, 8, 245, 12, 21, 16, 1089, 15, 1859, 21, 39, 28, 4335, 24, 6137, 24, 57, 40, 11109, 33, 115, 39, 45, 52, 22707, 35, 27869, 40, 93, 64, 55, 51, 47915, 57, 111, 76, 65559, 48, 75809, 56, 129, 88, 99405, 69, 329, 60, 119, 65, 143259, 72, 265, 63, 95, 112, 198417, 87, 219539

Offset: 3

N. J. A. Sloane, Apr 01 2013

In other words, k divides (reversal of k in base n), and (k-reversed)/k > 1.
The numbers are written in base 10.
Theorem: The length of k (in base n) is 2 iff n>=5 and n+1 is composite, otherwise 4.

			The numbers a(n) for n = 3, ..., 11 written in base n are 1012, 1023, 13, 1045, 15, 25, 17, 1089, 14.
For example, 1012 (base 3) = 32 (base 10), and 2101 (base 3) = 64 (base 10) = 2*32.

N. J. A. Sloane, paper in preparation.
See A214927 for further references and links.

Michel Marcus, Table of n, a(n) for n = 3..400
T. J. Kaczynski, Note on a Problem of Alan Sutcliffe, Math. Mag., 41 (1968), 84-86.
Lara Pudwell, Digit Reversal Without Apology, Mathematics Magazine, Vol. 80 (2007), pp. 129-132. Also arXiv:math/0511366 [math.HO], 2005.
Alan Sutcliffe, Integers That Are Multiplied When Their Digits Are Reversed, Mathematics Magazine, 39 (1966), 282-287.

Cf. A214927, A006093, A072668, A152619.

Table[k = 2; While[Nand[IntegerQ@ #, # != 1] &[FromDigits[#, n]/k] &@ Reverse@ IntegerDigits[k, n], k++]; k, {n, 3, 60}] (* Michael De Vlieger, Feb 26 2017 *)

isok(k, n) = {my(rk = fromdigits(Vecrev(digits(k, n)), n)); !(rk % k) && (rk > k);}
a(n) = {my(k = 1); while (!isok(k, n), k++); k;} \\ Michel Marcus, Feb 26 2017

If n=3 or n>3 and n+1 is prime, a(n) = (n^2-1)(n+1) (cf. A152619).

A173388 a(n) = a(n - 3) + a(n - 4) if n is even, else a(n - 2) + a(n - 3).

Original entry on oeis.org

1, 1, 1, 1, 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

Offset: 0

Roger L. Bagula, Feb 17 2010

The sequence is designed so that all the numbers, see A055389 or A006355, except the first two appear twice.
The limiting ratio a(n+1)/a(n) alternates between the golden ratio and one.
Essentially the same as A214927 (except for the first 4 terms and the offset). - Georg Fischer, Oct 01 2018

a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1;
a[n_] := a[n] = If[Mod[n, 2] == 0, a[n - 3] + a[n - 4], a[n - 2] + a[n - 3]];
Table[a[n], {n, 0, 50}]

a(n) = a(n-2)+a(n-4), n > 7.

G.f.: (x^7+1+x-x^6)/(1-x^2-x^4).

Formula added by the Assoc. Editors of the OEIS, Feb 24 2010

A223090 Numbers k with the property that if the base-8 representation of k is read backwards, the result is an integral multiple of k.

Original entry on oeis.org

21, 189, 525, 567, 1134, 1365, 1533, 4599, 4725, 9198, 10773, 12285, 34125, 36855, 38325, 73710, 86037, 87381, 96957, 98301, 269325, 294903, 307125, 589806, 688149, 700245, 774333, 786429, 2150925, 2184525, 2322999, 2359287, 2423925, 2457525, 4645998, 4718574, 5505045, 5515797, 5592405, 5603157, 6193341, 6204093, 6280701, 6291453

Offset: 1

N. J. A. Sloane, Mar 23 2013

A217742 is a subsequence.

The multiples that occur are 2, 3, 5, 7 (cf. A222817-A222820).

			n = 525 (base 10) = 1015 (base 8). Reading this backwards we get 5101 (base 8) = 2625 (base 10) = 5*n, so 525 is in the sequence.

Cf. A214927, A217742, A223089, A222817-A222820.

b8rQ[n_]:=Module[{c=FromDigits[Reverse[IntegerDigits[n,8]],8]},c/n>1 && IntegerQ[c/n]]; Select[Range[63*10^5],b8rQ] (* Harvey P. Dale, Sep 18 2016 *)

cp's OEIS Frontend

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.

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A222816 Numbers k that divide the base-5 reversal of k.

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A223078 Positive integers with the property that if the base-4 representation is reversed the result is three times the original number.

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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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A223089 Numbers n, written in base 8, with the property that if the base-8 representation of n is read backwards, the result is 5*n.

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A223079 Positive integers, written in base 4, with the property that if the base-4 representation is reversed the result is three times the original number.

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A224220 a(n) = smallest number k with property that if the base-n expansion of k is reversed, the result is a nontrivial multiple of k.

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A173388 a(n) = a(n - 3) + a(n - 4) if n is even, else a(n - 2) + a(n - 3).

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