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
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
CROSSREFS
KEYWORD
bref,hard,more,nonn
AUTHOR
Markus Sigg, Jul 15 2022
STATUS
approved