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A213340
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Numbers which are the values of the quadratic polynomial 5+8t+12k+16kt on nonnegative integers.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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EXAMPLE
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For n=5 the a(5)=29 solutions are {k=0, t=3}, {k=2, t=0}.
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MAPLE
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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;
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CROSSREFS
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Cf. A001844 (centered square numbers: 2n(n+1)+1).
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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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