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A272671 Numbers k such that the decimal number 1k is a square.

%I A272671 #36 Jul 20 2025 15:38:52
%S A272671 6,21,44,69,96,156,225,296,369,444,521,600,681,764,849,936,1025,1236,
%T A272671 1449,1664,1881,2100,2321,2544,2769,2996,3225,3456,3689,3924,4161,
%U A272671 4400,4641,4884,5129,5376,5625,5876,6129,6384,6641,6900,7161,7424,7689,7956,8225,8496,8769,9044,9321,9600
%N A272671 Numbers k such that the decimal number 1k is a square.
%H A272671 Davin Park, <a href="/A272671/b272671.txt">Table of n, a(n) for n = 1..20000</a>
%e A272671 44 is a member because 144 = 12^2 is a square.
%e A272671 0 is not a member because 10 is not a square.
%p A272671 t1:=[];
%p A272671 for k from 1 to 20000 do
%p A272671 if issqr(k+10^length(k)) then t1:=[op(t1),k]; fi;
%p A272671 od;
%p A272671 t1;
%t A272671 Flatten[n /. Solve[10^# + n == a^2 && 10^(# - 1) <= n < 10^# && a > 0, {n, a}, Integers] & /@ Range[3]] (* _Davin Park_, Feb 05 2017 *)
%t A272671 Select[Range[10000],IntegerQ[Sqrt[10^IntegerLength[#]+#]]&] (* _Harvey P. Dale_, Jul 20 2025 *)
%o A272671 (Magma) [n: n in [1..10000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1)))]; // _Marius A. Burtea_, Mar 21 2019
%o A272671 (PARI) isok(n) = issquare(eval(concat(1, Str(n)))); \\ _Michel Marcus_, Mar 21 2019
%o A272671 (Python)
%o A272671 from sympy.ntheory.primetest import is_square
%o A272671 def ok(n): return is_square(int('1'+str(n)))
%o A272671 print(list(filter(ok, range(9601)))) # _Michael S. Branicky_, Jun 21 2021
%Y A272671 Cf. A265432, A272672, A045855 (squares beginning with 1), A272684, A272685.
%K A272671 nonn,base
%O A272671 1,1
%A A272671 _Nathan Fox_, _Brooke Logan_, and _N. J. A. Sloane_, May 20 2016
%E A272671 Extended by _Davin Park_, Feb 05 2017