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

Author

Labos Elemer, Jun 03 2002

Keywords

• base
• easy
• nonn

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

Examples

			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.
		

Links

• 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]

Crossrefs

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

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

Programs

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 *)

Formula

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

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