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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.
2
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
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
KEYWORD
nonn
STATUS
approved