A275234 Number of distinct positive solutions to the system of n Diophantine equations: x_1 + y_1 = x_2*y_2, x_2 + y_2 = x_3*y_3, ..., x_n + y_n = x_1*y_1.

1, 2, 2, 4, 3, 6, 5, 10, 11, 17, 19, 36, 42, 70, 97, 155, 219, 351, 514, 815, 1228, 1918, 2937, 4614, 7111

Offset

1,2

Comments

In any solution, interchanging x_i and y_i for any i yields a new solution. So does a circular permutation of the solution. Two solutions are counted as distinct if one cannot be obtained from the other by these transformations.

Links

Table of n, a(n) for n=1..25.

Christopher Briggs, Python script for generating n-th term [warning: this program has errors]

Christopher Briggs, Y. Hirano, and H. Tsutsui, Positive Solutions to Some Systems of Diophantine Equations, Journal of Integer Sequences, 2016 Vol 19 #16.8.4.

Kinga Pósán and Szabolcs Tengely, SageMath code

Kinga Pósán, Table of solutions for n = 1..8

Example

For n = 1, the only positive solution to x + y = xy is x = y = 2.
For n = 2, the only distinct (see comments) positive solutions to x_1 + y_1 = x_2*y_2, x_2 + y_2 = x_1 * y_1 are (x_1,y_1,x_2,y_2) = (2,2,2,2) and (1,5,2,3).

Crossrefs

Sequence in context: A341465 A304406 A053197 * A301768 A088145 A011754

Adjacent sequences: A275231 A275232 A275233 * A275235 A275236 A275237

Keyword

nonn,more

Author

Christopher Briggs, Jul 20 2016

Extensions

Corrected by Kinga Pósán and Szabolcs Tengely, Mar 26 2022

Status

approved