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A213340 Numbers which are the values of the quadratic polynomial 5+8t+12k+16kt on nonnegative integers. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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,p pp 141-150.

LINKS

Table of n, a(n) for n=1..46.

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, Twin Pythagorean Triple (MathWorld).

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

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;

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.

Sequence in context: A166409 A077425 A039955 * A014539 A249034 A208883

Adjacent sequences:  A213337 A213338 A213339 * A213341 A213342 A213343

KEYWORD

nonn

AUTHOR

Michel Mizony, Jun 09 2012

STATUS

approved

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Last modified September 22 16:41 EDT 2019. Contains 327311 sequences. (Running on oeis4.)