reducible condition ?

[tommy1729]

let a_n be a strictly rising positive integer sequence.

a_n satisfies the following conditions and the question is if the conditions are reducible to a single condition.

there is also an unspecified strictly non-decreasing positive integer function f(n).

constant(k) means a constant depending on the positive integer k.
if constant(k) occurs twice it may be denoting two different functions resp constants.

sum n = 1 .. inf 1/a_n = oo

sum n = 1 .. inf 1/f(a_n) = oo

sum n = 1 .. inf f(a_n)^k /a_n - constant(k)/a_n = constant(k)

a_n > f(n).

f(n) > sum n = 1 .. n 1/a_(a_n)


... if that is even possible ...

regards

tommy1729