

A229941


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


2



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
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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 4term 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, 193 (2013), 2832.
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

JeanFrançois Alcover, Oct 04 2013


STATUS

approved



