Measurement of integrated luminosities at BESIII for data samples at center-of-mass energies between 4.0 and 4.6 GeV*Supported in part by National Key R&D Program of China (2020YFA0406300, 2020YFA0406400); National Natural Science Foundation of China (NSFC) (11625523, 11635010, 11735014, 11822506, 11835012, 11935015, 11935016, 11935018, 11961141012, 12022510, 12025502, 12035009, 12035013, 12061131003); the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS (U1732263, U1832207); CAS Key Research Program of Frontier Sciences (QYZDJ-SSW-SLH040); 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; ERC (758462); European Union Horizon 2020 research and innovation programme (Marie Sklodowska-Curie grant agreement No 894790); German Research Foundation DFG (443159800), Collaborative Research Center CRC 1044, GRK 2149; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey (DPT2006K-120470); National Science and Technology fund; Olle Engkvist Foundation (200-0605); STFC (United Kingdom); The Knut and Alice Wallenberg Foundation (Sweden) (2016.0157); The Royal Society, UK (DH140054, DH160214); The Swedish Research Council; U. S. Department of Energy (DE-FG02-05ER41374, DE-SC-0012069). (1st November 2022)