A051882 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.

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, 29, 33, 34, 35, 36, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 56, 58, 63, 68, 70, 72, 77

Offset: 1

Author

Jud McCranie, Dec 15 1999

Keywords

• nonn
• fini
• full
• nice

Comments

Lehmer shows that 77 is in this sequence. Graham shows that it is the last member of the sequence.

Examples

			1=1/2+1/3+1/6, so 2+3+6=11 is strict-sense Egyptian.
		

References

• D. H. Lehmer, unpublished work, cited in Graham 1963.
• See also R. K. Guy, Unsolved Problems Number Theory, Sect. D11.

Links

• R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3:4 (1963), pp. 435-441. [Alternate link]
• Eric Weisstein's World of Mathematics, Egyptian Number
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A028229.

Programs

Mathematica

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 *)