A087057 Smallest number whose square is larger than 2*n^2.
2, 3, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 68, 70, 71, 73, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 92, 94, 95, 97, 98, 99, 101
Offset: 1
Examples
a(10) = 15 because the 15^2 = 225 is the smallest square number greater than 2*10^2 = 200. Can be built by recursive removals: start with 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... get a(1) := 2 and remove the 2nd term (= 4): [2] _ 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ... get a(2) := 3 and remove the 3rd term (= 7): [2, 3] _ 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ... get a(3) := 5 and remove the 5th term (= 11): [2, 3, 5] _ 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ... get a(4) := 6 and remove the 6th term (= 14): [2, 3, 5, 6] _ 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, ... get a(5) := 8 and remove the 8th term (= 18): [2, 3, 5, 6, 8] _ 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, ... get a(6) = 9 and remove the 9th term (= 21), etc. - _Reinhard Zumkeller_, Feb 04 2014
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Programs
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Haskell
a087057 n = a087057_list !! (n-1) a087057_list = f [2..] where f (x:xs) = x : f (us ++ vs) where (us, _ : vs) = splitAt (x - 1) xs -- Reinhard Zumkeller, Feb 04 2014
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Mathematica
Ceiling[Range[110]Sqrt[2]] (* Harvey P. Dale, Oct 30 2013 *)
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PARI
a(n)=ceil(n*sqrt(2)) \\ Charles R Greathouse IV, Oct 24 2011
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PARI
a(n)=sqrtint(2*n^2+sqrtint(8*n^2)+1) \\ Charles R Greathouse IV, Oct 24 2011
Formula
a(n) = ceiling(n*sqrt(2)). - Vincenzo Librandi, Oct 22 2011
Comments