A177928 - OEIS

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.

%I #14 Jun 12 2017 17:32:14

%S 1,2,3,9,27,33,66,99,123,246,271,333,351,407,429,462,481,518,546,567,

%T 666,693,702,715,777,814,819,924,936,999,1434,2151,2868,3333,4521,

%U 4818,6666,7227,7373,7535,8631,9042,9999,33333,53658,54546,66666,80487,81819

%N 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.

%C A178028 is a subsequence of this sequence.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunits</a>

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

%p 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:

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

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

%K nonn,base

%O 1,2

%A _Michel Lagneau_, May 15 2010