A178028 Numbers n dividing every cyclic permutation of n^2.

1, 2, 3, 9, 27, 33, 99, 123, 271, 333, 351, 407, 429, 481, 693, 777, 819, 999, 2151, 3333, 4521, 7227, 7373, 9999, 33333, 81819, 99999, 194841, 326733, 333333, 340067, 366337, 369963, 386139, 389961, 437229, 534391, 623763, 706293, 762377, 863247

Offset: 1

Author

Michel Lagneau, May 17 2010

Keywords

• nonn
• base

Examples

			123 is a member as all the five cyclic permutations of 123^2 are :
{15129, 51291, 12915, 29151, 91512};
15129 = 123*123 ;
51291 = 123*417 ;
12915 = 123*105 ;
29151 = 123*237 ;
91512 = 123*744.
		

Crossrefs

Cf. A177950, A177928.

Programs

Maple

with(numtheory):for n from 1 to 100000 do:n0:=n^2:l:=length(n0) :ind:=0:for j
from 1 to l do:s:=0:for m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v
:s:=s+ u*10^m:od:s:=floor(s-u*10^l+u):if irem(s, n)=0 then ind:=ind+1:n0:=s:else
fi: od:if ind=l then printf(`%d, `, n):else fi: od:

Mathematica

Select[Range[900000],And@@Divisible[FromDigits/@Table[ RotateRight[ IntegerDigits[ #^2], n],{n,IntegerLength[#^2]}],#]&] (* Harvey P. Dale, Jul 31 2013 *)

Sage

def cycle(x): return (cp(x) for cp in CyclicPermutationGroup(len(x)))
is_A178028 = lambda n: all(n.divides(Integer(cx,base=10)) for cx in cycle(str(n**2))) # D. S. McNeil, Jan 08 2011