Direct Measurement of the Branching Fractions <math><mrow><mi>B</mi><mo>(</mo><mi>ψ</mi><mo>(</mo><mn>3686</mn><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>X</mi><mo>)</mo></mrow></math> and <math><mrow><mi>B</mi><mo>(</mo><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>X</mi><mo>)</mo></mrow></math>, and Observation of the State <math><mrow><mi>R</mi><mo>(</mo><mn>3760</mn><mo>)</mo></mrow></math> in <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>X</mi></mrow></math>

We report a measurement of the observed cross sections of e+e−→J/ψX based on 3.21 fb−1 of data accumulated at energies from 3.645 to 3.891 GeV with the BESIII detector operated at the BEPCII collider. In analysis of the cross sections, we measured the decay branching fractions of B(ψ(3686)→J/ψX)=(64.4&#177;0.6&#177;1.6)% and B(ψ(3770)→J/ψX)=(0.5&#177;0.2&#177;0.1)% for the first time. The energy-dependent line shape of these cross sections cannot be well described by two Breit-Wigner (BW) amplitudes of the expected decays ψ(3686)→J/ψX and ψ(3770)→J/ψX. Instead, it can be better described with one more BW amplitude of the decay R(3760)→J/ψX. Under this assumption, we extracted the R(3760) mass MR(3760)=3766.2&#177;3.8&#177;0.4 MeV/c2 , total width ΓR(3760)tot=22.2&#177;5.9&#177;1.4 MeV, and product of leptonic width and decay branching fraction ΓR(3760)eeB[R(3760)→J/ψX]=(79.4&#177;85.5&#177;11.7) eV. The significance of the R(3760) is 5.3σ. The first uncertainties of these measured quantities are from fits to the cross sections and second systematic.