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.

1, 1, 2, 6, 9, 16, 20, 30, 42, 49, 64, 81, 90, 110, 132, 156, 169, 196, 225, 256, 272, 306, 342, 380, 420, 441, 484, 529, 576, 625, 650, 702, 756, 812, 870, 930, 961, 1024, 1089, 1156, 1225, 1296, 1332, 1406, 1482, 1560, 1640

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

For n > 1: A257213(a(n)) = n. - Reinhard Zumkeller, Apr 19 2015

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A257213.

Programs

Haskell

a043548 n = f 1 where
   f k = if distinct $ (map (div k)) [n, n-1 .. 1] then k else f (k + 1)
   distinct [] = True; distinct (u:vs@(v:)) = u /= v && distinct vs
-- Reinhard Zumkeller, Apr 19 2015

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

Formula

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

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