A229941 Sequence of triples: the 10 solutions of 1/p + 1/q + 1/r = 1/2 with 0 < p <= q <= r, lexicographically sorted.

3, 7, 42, 3, 8, 24, 3, 9, 18, 3, 10, 15, 3, 12, 12, 4, 5, 20, 4, 6, 12, 4, 8, 8, 5, 5, 10, 6, 6, 6

Offset: 1

Author

Jean-François Alcover, Oct 04 2013

Keywords

• easy
• fini
• nonn
• full
• tabf

Comments

As noted by John Baez, "each of [the 10 solutions of 1/p + 1/q + 1/r = 1/2] gives a way for three regular polygons to snugly meet at a point".

Among the 14 4-term Egyptian fractions with unit sum, there are 10 of the form 1/2 + 1/p + 1/q + 1/r.

Also integer values of length, width and height of a rectangular prism whose surface area is equal to its volume: pqr = 2(pq+pr+qr). - John Rafael M. Antalan, Jul 05 2015

Examples

			a(1) = 3, a(2) = 7, a(3) = 42, since 1/3 + 1/7 + 1/42 = 1/2.
The 10 solutions are:
3,  7, 42;
3,  8, 24;
3,  9, 18;
3, 10, 15;
3, 12, 12;
4,  5, 20;
4,  6, 12;
4,  8,  8;
5,  5, 10;
6,  6,  6
		

Links

• John Baez, The answer is 42.
• J. F. T. Rabago and R. P. Tagle, On the Area and Volume of a certain Rectangular Solid and the Diophantine Equation 1/2=1/x+1/y+1/z, Notes on Number Theory and Discrete Mathematics, 19-3 (2013), 28-32.
• Wikipedia, Hurwitz's automorphisms theorem.

Crossrefs

Cf. A230400, A260819.

Programs

Mathematica

{p, q, r} /. {ToRules[Reduce[0 < p <= q <= r && 1/p + 1/q + 1/r == 1/2, {p, q, r}, Integers] ]} // Flatten