%I #18 Jul 30 2012 05:28:52
%S 2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T 29,33,34,35,36,39,40,41,42,44,46,47,48,49,51,56,58,63,68,70,72,77
%N Call m strict-sense Egyptian if we can partition m = x_1+x_2+...+x_k into distinct positive integers x_i such that Sum_{i=1..k} 1/x_i = 1; sequence gives all numbers that are not strict-sense Egyptian.
%C Lehmer shows that 77 is in this sequence. Graham shows that it is the last member of the sequence.
%D D. H. Lehmer, unpublished work, cited in Graham 1963.
%D See also R. K. Guy, Unsolved Problems Number Theory, Sect. D11.
%H R. L. Graham, <a href="http://www.math.ucsd.edu/~ronspubs/63_02_partitions.pdf">A theorem on partitions</a>, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441. [<a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4997964">Alternate link</a>]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EgyptianNumber.html">Egyptian Number</a>
%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e 1=1/2+1/3+1/6, so 2+3+6=11 is strict-sense Egyptian.
%t strictEgyptianQ[m_] := Length[ Select[ IntegerPartitions[m, Ceiling[(Sqrt[8*m + 1] - 1)/2]], Length[#] == Length[ Union[#]] && 1 == Plus @@ (1/#) & , 1]] > 0; Reap[ Do[ If[ !strictEgyptianQ[m], Print[m]; Sow[m]], {m, 1, 77}]][[2, 1]] (* _Jean-François Alcover_, Jul 30 2012 *)
%Y Cf. A028229.
%K nonn,fini,full,nice
%O 1,1
%A _Jud McCranie_, Dec 15 1999