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

1,1

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

Links

Table of n, a(n) for n=1..30.

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.

Example

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

Mathematica

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

Crossrefs

Cf. A230400, A260819.

Sequence in context: A179907 A080581 A086397 * A019018 A018993 A217759

Adjacent sequences: A229938 A229939 A229940 * A229942 A229943 A229944

Keyword

easy,fini,nonn,full,tabf

Author

Jean-François Alcover, Oct 04 2013

Status

approved