A355626 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that Product_{i = 1..n} (3 + 1/t_i) is integer.

0, 3, 80, 15222

Offset

1,2

Comments

Bounds on the values t_i can be derived as shown in "Bounds on t_i when the product of factors (3 + 1/t_i) is given" in the Links section. These bounds allow to calculate a(3) = 80 and a(4) = 15222 with the PARI program in the link. It seems that determining a(5) would need stronger methods. The identity (3 + 1/82) * (3 + 1/6670) * (3 + 1/44484454) * (3 + 1/1978866618021814) * (3 + 1/3915913091921090597566167836053) = 244 shows that the values t_n can get quite large. Integer products of more factors (3 + 1/t_i) can have even much larger t_n.

Links

Table of n, a(n) for n=1..4.

Markus Sigg, Bounds on t_i when the product of factors (3 + 1/t_i) is given

Markus Sigg, PARI program.

Example

a(1) = 0: As 1/t_1 is not integer for t_1 >= 2, there is no t_1 >= 2 with integer 3 + 1/t_1.
a(2) = 3: With p := (3 + 1/t_1) * (3 + 1/t_2) we have p > 9, so for integer p it is p >= 10. With p <= (3 + 1/t_1)^2 we get t_1 <= 6. Solving p = 10, p = 11, p = 12 with 2 <= t_1 <= 6 for t_2 shows that the only integer solutions are (t_1,t_2) = (4,13) and (t_1,t_2) = (5,8) for p = 10, and (t_1,t_2) = (2,7) for p = 11.

Prog

(PARI) See link. The program has code for sequences A355626, ..., A355631 and can handle the more general case of factors (s + 1/t_i) with s >= 2.

Crossrefs

Cf. A355627, A355628, A355629, A355630, A355631.

Sequence in context: A183265 A233124 A251694 * A274567 A137994 A074386

Adjacent sequences: A355623 A355624 A355625 * A355627 A355628 A355629

Keyword

bref,hard,more,nonn

Author

Markus Sigg, Jul 15 2022

Status

approved