One early snapshot of public opinion would suggest that Mr. Cain’s candidacy has not been diminished in the eyes of voters. A Washington Post/ABC News poll on Friday found that 7 in 10 Republicans say the allegations do not matter when it comes to selecting a presidential candidate.

The controversy around Mr. Cain, which intensified as the week wore on, is not likely to subside as long as differing accounts keep emerging. In a radio interview Thursday, Mr. Cain said, “How many more ways can I say this stuff is totally fabricated?”

His response to the allegations, first reported on Sunday by Politico, has provided the biggest test yet for an unorthodox campaign and a candidate whose popularity has been rooted in his unconventional style and outsider appeal to voters.

But his denials — he said repeatedly that his only crime had been telling a woman that she was as tall as his wife — have also fueled a sense of outrage among his accusers and their friends that he was using his national platform as a presidential candidate to rewrite history and diminish the seriousness of his actions.

As the week ended, Mr. Cain faced a continued stream of allegations, including new details about a different accuser, who, according to people familiar with the events, has said that he persistently propositioned her during a work event, and that the environment at the restaurant association turned hostile after she complained about it. That woman received a negotiated severance of $35,000 that included a confidentiality clause, people familiar with the agreement said. But it remains an open question how long the spectacle will hang over the presidential race. The candidates gather Wednesday in Michigan for a debate on the economy.

In Washington, Mr. Cain was met with a standing ovation by the crowd at an event hosted by Americans for Prosperity, a conservative group; some in the crowd even danced to his campaign theme song, “I Am America,” by Krista Branch. Mr. Cain once worked as a spokesman and promoter for the group, founded with support from the conservative billionaire industrialists Charles and David H. Koch.

Those ties were outlined in an article in The New York Times on Friday. Referring to that article, Mr. Cain declared: “I am the Koch brothers’ brother from another mother. Yes, I’m their brother from another mother, and proud of it.” As he said so, David Koch, beaming, stood up and pumped his fists in the air.