A273359 Numbers k such that the decimal number concat(4,k) is a square.
9, 41, 84, 225, 356, 489, 624, 761, 900, 1209, 1616, 2025, 2436, 2849, 3264, 3681, 4100, 4521, 4944, 5369, 5796, 6225, 6656, 7089, 7524, 7961, 8400, 8841, 9284, 9729, 10881, 12164, 13449, 14736, 16025, 17316, 18609, 19904, 21201, 22500
Offset: 1
Examples
84 is a member because 484 = 22^2 is a square. 0 is not a member because 40 is not a square. sqrt(410) < 21 AND 22 < sqrt(500) < 23 so 21^2 = 441 and 22^2 = 484 give 41 and 84 respectively. 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
Links
- Nathan Fox, Table of n, a(n) for n = 1..10000
Programs
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Magma
[n: n in [1..50000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(4)))]; // Vincenzo Librandi, Feb 20 2020
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Maple
t1:=[]; for k from 1 to 30000 do if issqr(k+4*10^length(k)) then t1:=[op(t1), k]; fi; od; t1;
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Mathematica
Select[Range[45000], IntegerQ[Sqrt[4 10^IntegerLength[#] + #]] &] (* Vincenzo Librandi, Feb 20 2020 *) DeleteCases[(FromDigits[Drop[IntegerDigits[#], 1]]) & /@ Select[Range[3, 500]^2, IntegerDigits[#][[1]] == 4 && IntegerDigits[#][[2]] != 0 &], 0] (* Alonso del Arte, Feb 20 2020 *)
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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 -
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
Comments