A213686 Numbers which are the values of the quadratic polynomial 13+20*t+24*k+32*k*t at nonnegative integers.

13, 33, 37, 53, 61, 73, 85, 89, 93, 109, 113, 133, 141, 145, 153, 157, 173, 181, 193, 201, 205, 213, 229, 233, 245, 253, 257, 273, 277, 293, 297, 301, 313, 317, 325, 333, 349, 353, 369, 373, 393, 397, 401, 405, 413, 421, 425, 433, 445, 453, 469, 473, 481

Offset

1,1

Comments

For all these numbers a(n) we have the following Erdos-Straus decomposition: 4/p=4/(13+24*k+20*t+32*k*t) = 1/(6*k+8*k*t+4+6*t) + 1/((13+24*k+20*t+32*k*t)*(5+8*k)*(3*k+4*k*t+2+3*t)) + 1/(2*(5+8*k)*(3*k+4*k*t+2+3*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+4k.

In 1948 Erdos 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).

References

I. Gueye and M. Mizony : Recent progress about Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2, pp. 6-14.

M. Mizony and I. Gueye : Towards the proof of Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2,p pp 141-150.

Links

Table of n, a(n) for n=1..53.

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 W. Weisstein, MathWorld: 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.

Example

For n=12 the a(12)=133  solutions are {k = 0, t = 6},{k = 5, t = 0}.

Maple

G:=(n, p)->4/p = [2*(2*n+1)/(n*p+p+1), 4/p/(n*p+p+1), 2/(n*p+p+1)]:
cousin:=proc(p)
local n;
for n from 0 to 300 do
if n*p+p+1 mod 4*(2*n+1)=0 then return([p, n, G(n, p)]); fi:
od:
end:
L:=NULL:for m to 400 do L:=L, cousin(4*m+1): od:{L}[1..4]; map(u->op(1, u), {L});

Crossrefs

Cf. A213340 (the quadratic polynomial 5+8t+12k+16kt).

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

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

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

Sequence in context: A282721 A206345 A284862 * A050659 A123161 A146052

Adjacent sequences: A213683 A213684 A213685 * A213687 A213688 A213689

Keyword

nonn

Author

Michel Mizony, Jun 17 2012

Status

approved