A213340 Numbers which are the values of the quadratic polynomial 5+8t+12k+16kt on nonnegative integers.

5, 13, 17, 21, 29, 37, 41, 45, 53, 61, 65, 69, 77, 85, 89, 93, 97, 101, 109, 113, 117, 125, 133, 137, 141, 149, 153, 157, 161, 165, 173, 181, 185, 189, 197, 205, 209, 213, 221, 229, 233, 237, 241, 245, 253, 257

Offset: 1

Author

Michel Mizony, Jun 09 2012

Keywords

• nonn

Comments

For all these numbers a(n) we have the following Erdős-Straus decomposition: 4/p = 4/(5+8*t+12*k+16*k*t) = 1/(2*(2*k+1)*(2+3*t+3*k+4*k*t)) + 1/(2+3*t+3*k+4*k*t) + 1/(2*(5+8*t+12*k+16*k*t)*(2*k+1)*(2+3*t+3*k+4*k*t)).

Moreover this sequence is related to irreducible twin Pythagorean triples: the associated Pythagorean triple is [2n(n+1), 2n+1,2n(n+1)+1], where n=2+4t+6k+8kt.

In 1948 Erdős and Straus conjectured that for any positive integer n >= 2 the equation 4/n = 1/x + 1/y +1/z has a solution with positive integers x, y and z (without the additional requirement 0 < x < y < z).

For the solution (x,y,z) having the largest z value, see (A075245, A075246, A075247).

Examples

			For n=5 the a(5)=29 solutions are {k=0, t=3}, {k=2, t=0}.
		

References

• I. Gueye and M. Mizony, Recent progress about Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2, pp. 6-14.
• I. Gueye and M. Mizony, Towards the proof of Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2, pp. 141-150.

Links

• P. Erdős, On a Diophantine equation, (Hungarian. Russian, English summaries), Mat. Lapok 1, 1950, pp. 192-210.
• M. Mizony and M.-L. Gardes, Sur la conjecture d'Erdős et Straus, see pages 14-17.
• Eric Weisstein's World of Mathematics, Twin Pythagorean Triple.
• K. Yamamoto, On the Diophantine Equation 4/n=1/x+1/y+1/z, Mem. Fac. Sci. Kyushu U. Ser. A 19, 37-47, 1965.

Crossrefs

Cf. A001844 (centered square numbers: 2n(n+1)+1).

Cf. A005408 (x values), A046092 (y values).

Cf. A195770 (positive integers a for which there is a 1-Pythagorean triple (a,b,c) satisfying a<=b).

A073101 number of solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.

Programs

Maple

G:=(p,d)->4/p = [4*d/(p+d)/(p+1), 4/(p+d), 4*d/(p+d)/(p+1)/p]:
cousin:=proc(p)
local d;
for d from 3 by 4 to 100 do
if ((p+1)/2) mod d=0 and (p+d)*(p+1) mod d=0 then
return([p,G(p,d)]) fi;od;
end:
for k to 20 do cousin(4*k+1) od;