How many $pep$-related electron-neutrinos $\nu_e$ does the Sun produce? In the sun neutrinos can be produced by the rare proton-electron-proton (pep) reaction:
$$ {}^1_1 H + e + {}^1_1 H \to {}^2_1 H + \nu_e $$
How many pep-related electron neutrinos does the Sun produce?
 A: The solar pep neutrinos were first detected in 2011 by Borexino collaboration (arXiv:1110.3230). The abstract reads:

We observed, for the first time, solar neutrinos in the 1.0-1.5 MeV energy range. We measured the rate of pep solar neutrino interactions in Borexino to be [3.1$\pm$0.6(stat)$\pm$0.3(syst)] counts/(day x 100 ton) and provided a constraint on the CNO solar neutrino interaction rate of < 7.9 counts/(day x 100 ton) (95% C.L.). The absence of the solar neutrino signal is disfavored at 99.97% C.L., while the absence of the pep signal is disfavored at 98% C.L. This unprecedented sensitivity was achieved by adopting novel data analysis techniques for the rejection of cosmogenic $^{11}$C, the dominant background in the 1-2 MeV region. Assuming the MSW-LMA solution to solar neutrino oscillations, these values correspond to solar neutrino fluxes of [1.6$\pm$0.3]$\times 10^8$ cm$^{-2}$s$^{-1}$ and 7.7$\times10^8$ cm$^{-2}$s$^{-1}$ (95% C.L.), respectively, in agreement with the Standard Solar Model. These results represent the first measurement of the pep neutrino flux and the strongest constraint of the CNO solar neutrino flux to date.

Since you wish to know the rate of pep neutrinos produced, we just need to multiply this flux with area of a sphere with a radius of 1 AU (Earth-Sun distance):
$$
\frac{dN}{dt} = \phi A = 1.6\times 10^{12}\: \mathrm{m}^{-2}\mathrm{s}^{-1} \times 4\pi\times (1.5 \times 10^{11} \:\text{m})^2 \approx \underline{4.5 \times 10^{35} \: \mathrm{s}^{-1}}
$$
So even though it took very long for us to detect them, the number of pep neutrinos created is quite large indeed!
