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A087057 Smallest number whose square is larger than 2*n^2. 8

%I

%S 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,

%T 37,39,40,42,43,44,46,47,49,50,51,53,54,56,57,58,60,61,63,64,66,67,68,

%U 70,71,73,74,75,77,78,80,81,83,84,85,87,88,90,91,92,94,95,97,98,99,101

%N Smallest number whose square is larger than 2*n^2.

%C Integer solutions to the equation x=ceiling(r*floor(x/r)) where r=sqrt(2). - _Benoit Cloitre_, Feb 14 2004

%H Reinhard Zumkeller, <a href="/A087057/b087057.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 + A001951(n) = 1 + floor(n*sqrt(2)) = sqrt(A087058(n)).

%F a(n) = ceiling(n*sqrt(2)). - _Vincenzo Librandi_, Oct 22 2011

%e a(10) = 15 because the 15^2 = 225 is the smallest square number greater than 2*10^2 = 200.

%e Can be built by recursive removals:

%e start with 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...

%e get a(1) := 2 and remove the 2nd term (= 4):

%e [2] _ 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...

%e get a(2) := 3 and remove the 3rd term (= 7):

%e [2, 3] _ 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, ...

%e get a(3) := 5 and remove the 5th term (= 11):

%e [2, 3, 5] _ 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, ...

%e get a(4) := 6 and remove the 6th term (= 14):

%e [2, 3, 5, 6] _ 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, ...

%e get a(5) := 8 and remove the 8th term (= 18):

%e [2, 3, 5, 6, 8] _ 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, ...

%e get a(6) = 9 and remove the 9th term (= 21), etc.

%e - _Reinhard Zumkeller_, Feb 04 2014

%t Ceiling[Range[110]Sqrt[2]] (* _Harvey P. Dale_, Oct 30 2013 *)

%o (PARI) a(n)=ceil(n*sqrt(2)) \\ _Charles R Greathouse IV_, Oct 24 2011

%o (PARI) a(n)=sqrtint(2*n^2+sqrtint(8*n^2)+1) \\ _Charles R Greathouse IV_, Oct 24 2011

%o (Haskell)

%o a087057 n = a087057_list !! (n-1)

%o a087057_list = f [2..] where

%o f (x:xs) = x : f (us ++ vs) where (us, _ : vs) = splitAt (x - 1) xs

%o -- _Reinhard Zumkeller_, Feb 04 2014

%Y Cf. A001951, A087055, A087056, A087058, A087059, A087060.

%K easy,nonn

%O 1,1

%A _Jens Voß_, Aug 07 2003

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Last modified July 24 15:05 EDT 2021. Contains 346273 sequences. (Running on oeis4.)