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Numbers n such that floor(sqrt(n)) = floor(n/floor(sqrt(n)))-1.
5

%I #38 Sep 08 2022 08:46:04

%S 2,6,7,12,13,14,20,21,22,23,30,31,32,33,34,42,43,44,45,46,47,56,57,58,

%T 59,60,61,62,72,73,74,75,76,77,78,79,90,91,92,93,94,95,96,97,98,110,

%U 111,112,113,114,115,116,117,118,119,132,133,134,135,136

%N Numbers n such that floor(sqrt(n)) = floor(n/floor(sqrt(n)))-1.

%C One of four sequences given by classifying natural numbers according to the value of floor(sqrt(n)). See the paper in Link lines and A005563, A217570, A217571.

%C Can be interpreted as a triangle read by rows: T(n,k) = n*(n+1)+k-1 with n>0, k=1..n. - _Bruno Berselli_, Oct 11 2012

%H Reinhard Zumkeller, <a href="/A217575/b217575.txt">Table of n, a(n) for n = 1..10000</a>

%H Takumi Sato, <a href="http://vixra.org/abs/1210.0025">Classification of Natural Numbers</a>

%F a(n) = A063657(n) - 1. - _Reinhard Zumkeller_, Jun 20 2015

%e As a triangle (see the second comment) this begins:

%e 2;

%e 6, 7;

%e 12, 13, 14;

%e 20, 21, 22, 23;

%e 30, 31, 32, 33, 34;

%e 42, 43, 44, 45, 46, 47;

%e 56, 57, 58, 59, 60, 61, 62;

%e 72, 73, 74, 75, 76, 77, 78, 79;

%e 90, 91, 92, 93, 94, 95, 96, 97, 98; etc.

%e - _Bruno Berselli_, Oct 11 2012

%t Select[Range[200],Floor[Sqrt[#]]==Floor[#/Floor[Sqrt[#]]]-1&] (* _Harvey P. Dale_, Oct 06 2018 *)

%o (Visual Basic in Excel)

%o Sub A217575()

%o Dim x As Long, n As Long, y As Long, i As Long

%o x = InputBox("Count to")

%o For n = 2 To x

%o y = Int(Sqr(n))

%o If y = Int(n / y) - 1 Then

%o i = i + 1

%o Cells(i, 1) = n

%o End If

%o Next n

%o End Sub

%o (Magma) [n: n in [1..150] | Isqrt(n) eq Floor(n/Isqrt(n))-1]; // _Bruno Berselli_, Oct 08 2012

%o (PARI) is_A217575(n)=n\(n=sqrtint(n))-1==n \\ - _M. F. Hasler_, Oct 09 2012

%o (Haskell)

%o a217575 = subtract 1 . a063657 -- _Reinhard Zumkeller_, Jun 20 2015

%Y Cf. A005563, A217570, A217571.

%Y Cf. A063657.

%K nonn

%O 1,1

%A _Takumi Sato_, Oct 07 2012