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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.

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

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OFFSET

1,3

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

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

PROG

(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