A213687 Numbers which are the values of the quadratic polynomial 3+4*k+7*t+8*k*t on nonnegative integers.

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

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

Table of n, a(n) for n=1..61.

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