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A260402 Numbers which cannot be the largest denominator of an Egyptian fraction for 1. 0

%I #20 Aug 07 2015 10:55:11

%S 2,3,4,5,7,8,9,10,11,13,14,16,17,19,21,22,23,25,26,27,29,31,32,34,37,

%T 38,39,41,43,44,46,47,49,50,51,53,57,58,59,61,62,64,67,68,69,71,73,74,

%U 79,81,82,83,86,87,89,92,93,94,97,98,101,103,106,107,109

%N Numbers which cannot be the largest denominator of an Egyptian fraction for 1.

%C Complement of A092671.

%C Contains at all primes and prime powers (A000961).

%C Martin studies the asymptotic behavior of this sequence: the order of magnitude of its counting function (number of elements below x) is x log log x / log x.

%H G. Martin, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa95/aa9533.pdf">Denser Egyptian fractions</a>, Acta Arith. 95 (2000), no. 3, 231-260.

%H G. Martin, <a href="http://www.math.ubc.ca/~gerg/slides/Urbana-27March09.pdf">Dense Egyptian fractions</a>, Talk at the AMS Spring Central Sectional Meeting, University of Illinois at Urbana-Champaign, March 27, 2009.

%e 10 is in this sequence because any Egyptian fraction with 1/10 as its term with largest denominator either contains 1/5 as well or not; either way, the resulting sum will have a factor 5 in its denominator (any other term will contribute a multiple of 5 to the numerator of the sum), hence cannot equal 1.

%K nonn

%O 1,1

%A _M. F. Hasler_, Jul 24 2015

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