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tokens: (pessimistic). The most pessimistic lower limit for the background is ≈1 × 10 −16 (Dvorkin Barausse 2017 ; Bonetti et al. 2018 ), though most estimates suggest that A SB will be at least several times larger (Ravi et al. 2015 ). We also looked at two different values of the total time span observed after the end of NG11, T + = 10 or 20 years, for calculating ρ . In Table 1 , we show our results in terms of the amount of time per pulsar per year allocated given different values of A SB and T + . Table 1 Optimal Pulsar Cadences for a Stochastic Background Note. We assumed β = 13/3. Added time span T + is in units of years. Dashed entries formally have values, but are too small to be reported. 3.1. Maintaining the PTA: The Time Allocation per Pulsar Table 1 shows that our best pulsars do not overwhelmingly dominate the stochastic-background time allocation. This makes sense when we consider a hypothetical PTA where we ignore all T ij and χ ij and only have pulsars with white noise. Following Siemens et al. ( 2013 ) but including the differences in white noise and cadences, we have . Then, one can easily show that the total ρ maximizes when the “quality” of each pulsar, the white-noise spectrum , is similar, and thus one should spend more time on pulsars with higher white-noise rms. Lee et al. ( 2012 ) reached a similar result. Since the optimization performed is over complex nonlinear equations (Equations ( 1 ), ( 2 )), the results we obtain for an individual pulsar can differ compared with that of the scaling relations. For real PTAs, one should consider red noise, varying time baselines, changes in the relative contributions of white-noise parameters over time, and sky positions, which will modify the results accordingly, though the basic principle still applies. We see that the time allocation is of similar magnitude for all pulsars, i.e., no pulsar overwhelmingly dominates the time allocation, though several pulsars do not contribute significantly for certain assumed parameters and thus are not listed (when c i 1 hr/year). Comparing against the value of ρ determined from a continuation of the NG11 observing strategy versus the optimizations found here, we see a ∼5% increase in ρ for , but a nearly ∼8%–10% increase for . NANOGrav’s general strategy of observing many high-precision pulsars with roughly equal time (along with high-cadence programs) therefore currently provides close to optimal stochastic-background sensitivity. 3.2. Trimming the PTA: Removing Pulsars to Improve Sensitivity Figure 1 shows the effect of removing a pulsar from the array and whether reallocating its observing time will increase or decrease sensitivity to the stochastic background. We find that all 45 pulsars contribute significantly for decades-long experiments and that the relative importance of continuing to time these pulsars also tends to improve. However, note in Table 1 where certain pulsars become significant for T + = 20 years, whereas they are unlisted for