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

1,2

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.

Links

Table of n, a(n) for n=1..49.

Eric Weisstein's World of Mathematics, Repunits

Example

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

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 *)

Crossrefs

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

Sequence in context: A096237 A380629 A309814 * A057231 A178028 A045596

Adjacent sequences: A177925 A177926 A177927 * A177929 A177930 A177931

Keyword

nonn,base

Author

Michel Lagneau, May 15 2010

Status

approved