A177928 Let n be the number whose square n^2 has the decimal expansion { d(1) d(2) ... d(D) }, and let q be the corresponding number whose decimal expansion is { d(2) d(3) ... d(D) d(1)}. Sequence lists numbers n dividing q.

1, 2, 3, 9, 27, 33, 66, 99, 123, 246, 271, 333, 351, 407, 429, 462, 481, 518, 546, 567, 666, 693, 702, 715, 777, 814, 819, 924, 936, 999, 1434, 2151, 2868, 3333, 4521, 4818, 6666, 7227, 7373, 7535, 8631, 9042, 9999, 33333, 53658, 54546, 66666, 80487, 81819

Offset: 1

Author

Michel Lagneau, May 15 2010

Keywords

• nonn
• base

Comments

A178028 is a subsequence of this sequence.

When n divides q, n divides d(D)*(10^D - 1) because q = 10*n^2 - d(D)*(10^D - 1). If n is prime, n divides (10^D - 1); for example, the prime term 271 divides 10^5 - 1 = 99999 = 271*369.

Examples

			429 is in the sequence because 429^2 = 184041 and 840411/429 = 1959.
		

Links

• Eric Weisstein's World of Mathematics, Repunits

Crossrefs

Cf. A002275, A003020, A005422, A067063, A102380, A178028.

Programs

Maple

for n from 1 to 10^6 do: d:=convert(n^2, base, 10):n1:=nops(d):s:=sum('d[i]*10^i','i'=1..n1-1)+d[n1]:if irem(s,n)=0 then printf(`%d, `,n):else fi:od:

Mathematica

Select[Range[100000], Mod[FromDigits[RotateLeft[IntegerDigits[#^2]]], #] == 0 &] (* T. D. Noe, Jul 27 2012 *)