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

1,3

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

Crossrefs

Cf. A257213.

Sequence in context: A320496 A172433 A049622 * A347535 A354975 A280228

Adjacent sequences: A043545 A043546 A043547 * A043549 A043550 A043551

Keyword

nonn

Author

Clark Kimberling

Status

approved