Vô conjecture 31 series 2(VC31S2)

We introduce a one-parameter family of multiplicative arithmetic functions (fc:ℕ→ℕ) defined by a shifted prime-power rule. We investigate additive representation properties of the image set of fc. In particular, we formulate a threshold phenomenon conjecture describing when sufficiently large integers can be expressed as sums of two values in the image of fc, and when such representations fail infinitely often.