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A230604 Smallest number whose square has more than n digits and begins and ends with the same n digits.

%I A230604 #67 Oct 14 2024 17:55:38
%S A230604 11,173,264,16262,193744,238165,38981039,112791955,1580178016,
%T A230604 1052631579,30762132977,15020242915,14451789007487,10909090909091,
%U A230604 1242844268897055,1001889106154509,4024018444782046,10018891061545090,11678332116788271168,102040816316530612245,139009056141395353279,1128182832632197435939
%N A230604 Smallest number whose square has more than n digits and begins and ends with the same n digits.
%C A230604 Differs from A161355 in that the present sequence allows an "overlap" of the digits, while A161355 requires a(n)^2 to have at least 2n digits. - _M. F. Hasler_, Mar 03 2014
%H A230604 Max Alekseyev, <a href="/A230604/b230604.txt">Table of n, a(n) for n = 1..32</a>
%e A230604 a(3)=264 since 264^2 = 69696 is the smallest square that starts and ends with the same 3 digits.
%t A230604 a[n_] := Block[{digits = {}},
%t A230604   For[i = Ceiling[Sqrt[10^n]], True, i++,
%t A230604    If[i^2 >= 10^n, digits = IntegerDigits[i^2];
%t A230604     If[Take[digits, n] == Take[digits, -n], Return[i]]]]];
%t A230604 a2[#] & /@ Range[1, 6] (* _Julien Kluge_, Feb 02 2016 *)
%o A230604 (PARI) for(n=1, 8, k=floor(sqrt(10^n)); until(bn==ed, k++; sr=Str(k^2); vc=Vec(sr); ln=#sr; bn=vc[1..n]; ed=vc[ln-n+1..ln]); print1(k, ", "));
%Y A230604 Cf. A116501, A161355.
%K A230604 nonn,base
%O A230604 1,1
%A A230604 _Arkadiusz Wesolowski_, Feb 28 2014
%E A230604 a(9)-a(10) from _Julien Kluge_, Feb 13 2016
%E A230604 a(11)-a(12) from _Julien Kluge_, Mar 04 2016
%E A230604 a(13) from _Giovanni Resta_, Apr 18 2016
%E A230604 Terms a(14) onward from _Max Alekseyev_, Oct 11 2024