A166320 Numbers which are proper divisors of the number obtained by cyclic right-shift by 2 positions.

142857, 153846, 190476, 230769, 238095, 307692, 142857142857, 153846153846, 190476190476, 230769230769, 238095238095, 307692307692, 1176470588235294, 100250626566416040, 102756892230576441, 105263157894736842, 107769423558897243

Offset: 1

Author

Claudio Meller, Oct 11 2009

Keywords

• nonn
• base

Comments

Numbers which are invariant under the cyclic shift are not in the sequence, because they are not proper divisors of themselves.

Examples

			153846 is a proper divisor of 461538, obtained by moving the two least significant digits "46" to the front.
230769 is a proper divisor of 692307, obtained by moving the least significant digits "69" to the front.
		

Programs

Maple

cycShft := proc(n) local L,Ls ; if n < 100 then n; else L := convert(n,base,10) ; Ls := [op(3..nops(L),L),op(1..2,L)] ; add(op(i,Ls)*10^(i-1),i=1..nops(Ls)) ; end if; end proc:
isA166320 := proc(n) local c; c := cycShft(n) ; return c mod n = 0 and c <> n ; end:
for n from 100 do if isA166320(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Oct 14 2009

PARI

{ genupto(m) = R = []; for(k=2,9, for(a=10*k,99, d = 100*k - 1; q = znorder( Mod(10, d/gcd(d,a)) ); forstep(z=q,m,q, R = concat(R,[(10^z-1)*a/d]); ); ); ); vecsort(R,,8) } \\ generate all terms of length up to m. Max Alekseyev, Jan 22 2012

Extensions

keyword:base added by R. J. Mathar, Oct 14 2009

Corrected my number typo in the examples - R. J. Mathar, Oct 23 2009

More terms from Sean A. Irvine, Feb 25 2010

More terms from Max Alekseyev, Jan 22 2012