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A273359 Numbers k such that the decimal number concat(4,k) is a square.

%I A273359 #24 Jan 17 2024 09:12:09
%S A273359 9,41,84,225,356,489,624,761,900,1209,1616,2025,2436,2849,3264,3681,
%T A273359 4100,4521,4944,5369,5796,6225,6656,7089,7524,7961,8400,8841,9284,
%U A273359 9729,10881,12164,13449,14736,16025,17316,18609,19904,21201,22500
%N A273359 Numbers k such that the decimal number concat(4,k) is a square.
%C A273359 Elements are squares of integers in (sqrt(41), sqrt(50)) * sqrt(10)^k without the leading 4 elements for nonnegative k. - _David A. Corneth_, May 20 2016
%H A273359 Nathan Fox, <a href="/A273359/b273359.txt">Table of n, a(n) for n = 1..10000</a>
%e A273359 84 is a member because 484 = 22^2 is a square.
%e A273359 0 is not a member because 40 is not a square.
%e A273359 sqrt(410) < 21 AND 22 < sqrt(500) < 23 so 21^2 = 441 and 22^2 = 484 give 41 and 84 respectively.
%e A273359 64 < sqrt(4100) < 65 AND 70 < sqrt(5000) < 71 so 65^2 = 4225, 66^2 = 4356, ..., 70^2 = 4900 give 225, 356, ..., 900 respectively. - _David A. Corneth_, May 20 2016
%p A273359 t1:=[];
%p A273359 for k from 1 to 30000 do
%p A273359 if issqr(k+4*10^length(k)) then t1:=[op(t1), k]; fi;
%p A273359 od;
%p A273359 t1;
%t A273359 Select[Range[45000], IntegerQ[Sqrt[4 10^IntegerLength[#] + #]] &] (* _Vincenzo Librandi_, Feb 20 2020 *)
%t A273359 DeleteCases[(FromDigits[Drop[IntegerDigits[#], 1]]) & /@ Select[Range[3, 500]^2, IntegerDigits[#][[1]] == 4 && IntegerDigits[#][[2]] != 0 &], 0] (* _Alonso del Arte_, Feb 20 2020 *)
%o A273359 (PARI) a(n) = {my(k=1,t=0); while(n>k, n-=k; t++; k=floor(sqrt(50)*sqrt(10^t))- ceil(sqrt(41)*sqrt(10^t))+1);(ceil(sqrt(41)*sqrt(10^t))+n-1)^2%(40*10^t)} \\ _David A. Corneth_, May 20 2016
%o A273359 (Magma) [n: n in [1..50000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(4)))]; // _Vincenzo Librandi_, Feb 20 2020
%o A273359 (Scala) (3 to 500).map(n => n * n).filter(n => n.toString.startsWith("4") && !n.toString.startsWith("40")).map(n => Integer.parseInt(n.toString.substring(1))) // _Alonso del Arte_, Feb 20 2020
%Y A273359 Cf. A272671, A273357, A273358, A273360, A273361, A273362, A273363, A273364.
%K A273359 nonn,base
%O A273359 1,1
%A A273359 _Nathan Fox_, _Brooke Logan_, and _N. J. A. Sloane_, May 20 2016