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A043548 Least separator of first n Egyptian fractions; i.e., least k for which the integers floor(k/m) for m=1,2,...,n are distinct.

%I A043548 #26 Feb 13 2023 02:33:29
%S A043548 1,1,2,6,9,16,20,30,42,49,64,81,90,110,132,156,169,196,225,256,272,
%T A043548 306,342,380,420,441,484,529,576,625,650,702,756,812,870,930,961,1024,
%U A043548 1089,1156,1225,1296,1332,1406,1482,1560,1640
%N A043548 Least separator of first n Egyptian fractions; i.e., least k for which the integers floor(k/m) for m=1,2,...,n are distinct.
%C A043548 For n > 1: A257213(a(n)) = n. - _Reinhard Zumkeller_, Apr 19 2015
%C A043548 After the initial 1, 1, the sequence appears to alternate between runs of pronic numbers and squares with run lengths 2,2,3,3,4,4,... - _Charlie Neder_, Oct 04 2018
%H A043548 Reinhard Zumkeller, <a href="/A043548/b043548.txt">Table of n, a(n) for n = 1..1000</a>
%F A043548 a(n) = n^2 + floor(sqrt(n-1))*floor(sqrt(n)+1/2) - n*floor(sqrt(n-1)) - n*floor(sqrt(n)+1/2), for n>1. - _Ridouane Oudra_, Jun 08 2020
%F A043548 a(n) = n^2 - n*t + floor((t^2)/4), where t = floor(sqrt(4*n-3)) for n>1. - _Ridouane Oudra_, Jan 24 2023
%o A043548 (Haskell)
%o A043548 a043548 n = f 1 where
%o A043548    f k = if distinct $ (map (div k)) [n, n-1 .. 1] then k else f (k + 1)
%o A043548    distinct [_] = True; distinct (u:vs@(v:_)) = u /= v && distinct vs
%o A043548 -- _Reinhard Zumkeller_, Apr 19 2015
%o A043548 (PARI) a(n)={if(n==1, 1, my(t=sqrtint(4*n-3)); n^2 - n*t + t^2\4)} \\ _Andrew Howroyd_, Feb 04 2023
%Y A043548 Cf. A257213.
%K A043548 nonn
%O A043548 1,3
%A A043548 _Clark Kimberling_