A108287 Consider the Diophantine equations x+y=a, x^2+y^2=b, x^3+y^3=c. There are three values of a for each pair b,c with b>0 and c>0. Sequence gives values of b.

1, 4, 7, 9, 13, 16, 19, 21, 25, 28, 31, 36, 37, 39, 43, 49, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 81, 84, 91, 91, 93, 97, 100, 103, 109, 111, 112, 117, 121, 124, 127, 129, 133, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 169, 171, 172, 175, 181, 183

Offset

1,2

Comments

From Andrew Howroyd, Mar 14 2023: (Start)

The following information was extracted from the Derive script and notes.

The values of x and y are not required to be integers or even real numbers.

Eliminating x and y from the equations gives:

a^3 - 3*a*b + 2*c = 0.

This is the Diophantine equation that is being solved. For a given b and c there will be three possibly complex roots. It is required to find those values of b and c where all three roots are integers.

The values for b are the integers of the form (d^2 + d*e + e^2)/3 where d and e are positive integers and d <= e. This sequence lists these values in order with repetition.

The corresponding values for c are d*e*(d+e)/2 given in A108940, and the three values for a are d, e and -(d+e).

(End)

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

James R FitzSimons, Derive program

Prog

(Derive) See Links section
(PARI) upto(blim)={my(L=List()); for(e=1, sqrtint(3*blim), for(d=1, e, my(b=(d^2+d*e+e^2)/3); if(b<=blim && !frac(b), listput(L, b)))); listsort(L); Vec(L)} \\ Andrew Howroyd, Mar 13 2023

Crossrefs

For values of c see A108940.

Sequence in context: A310960 A376208 A376210 * A230240 A243175 A352272

Adjacent sequences: A108284 A108285 A108286 * A108288 A108289 A108290

Keyword

nonn

Author

James R FitzSimons (cherry(AT)getnet.net), Jun 22 2007

Extensions

Terms a(42) and beyond from Andrew Howroyd, Mar 14 2023

Status

approved