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 A213687 Numbers which are the values of the quadratic polynomial 3+4*k+7*t+8*k*t on nonnegative integers. 0
 3, 7, 10, 11, 15, 17, 19, 22, 23, 24, 27, 31, 34, 35, 37, 38, 39, 43, 45, 46, 47, 51, 52, 55, 57, 58, 59, 63, 66, 67, 70, 71, 73, 75, 77, 79, 80, 82, 83, 87, 91, 94, 95, 97, 99, 101, 103, 106, 107, 108, 111, 112, 115, 117, 118, 119, 122, 123, 126, 127, 129 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For all these numbers a(n) we have the following Erdos-Straus decomposition:  4/p= 4/(3+4*k+7*t+8*k*t) = 1/(2*(3+4*k+7*t+8*k*t)*(1+k)) + 1/((1+k)*(2*t+1)) + 1/(2*(1+k)*(2*t+1)*(3+4*k+7*t+8*k*t))); Moreover this sequence is related to irreducible twin Pythagorean triples: the associated Pythagorean triple is [2t(t+1),2t+1, 2t(t+1)+1]. 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 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)=31  solutions are {k = 0, t = 4},{k = 7, t = 0}. MAPLE H:=(k, t) -> 4/(3+4*k+7*t+8*k*t) = [1/2*1/((3+4*k+7*t+8*k*t)*(1+k)), 1/((1+k)*(2*t+1)), 1/2*1/((1+k)*(2*t+1)*(3+4*k+7*t+8*k*t))]: cousin:=proc(p) local n, k; for n from 0 to (p-3)/7 do if (p-3-7*n) mod (4+8*n)=0  then k:=(p-3-7*n)/(4+8*n): return([p, n, H(k, n)]) fi; od; end: L:=NULL:for p from 2 to 500 do L:=L, cousin(p): od:{L}[1..10]; map(u->op(1, u), {L}); map(u->op(2, 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: A190677 A310177 A324774 * A226934 A238506 A308169 Adjacent sequences:  A213684 A213685 A213686 * A213688 A213689 A213690 KEYWORD nonn AUTHOR Michel Mizony, Jun 18 2012 STATUS approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)