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

1,1

Comments

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

Links

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

N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.

Formula

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

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A001232, A222814, A008918, A222815, A008919, A031877.

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

Sequence in context: A168661 A175698 A110819 * A008919 A110843 A354256

Adjacent sequences: A071682 A071683 A071684 * A071686 A071687 A071688

Keyword

base,easy,nonn

Author

Labos Elemer, Jun 03 2002

Extensions

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

Status

approved