WASHINGTON — President Trump, who has taken to referring to himself as “T” on Twitter, demanded on Monday that his critics give him “an apology” for any suggestion that “T people” have colluded with Russia or obstructed justice — continuing a weeklong attack on the Obama administration’s response to Moscow’s attempt to influence the 2016 election.

In a series of four tweets that he began sending shortly after 8 a.m., Mr. Trump also echoed a favorite theme from last week: that President Barack Obama, and not “T,” had failed to push back against Russian interference in the election.

Mr. Trump’s tweets were additional evidence of his refusal to acknowledge a basic fact agreed upon by 17 American intelligence agencies that he now oversees: that Russia orchestrated the hacking attacks, and did so to help get him elected.

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Many Republicans, including his aides, wish the president would stop focusing on the issue. But Mr. Trump sees questions about Russia’s interference in the election as an effort by Democrats and stragglers from the “Never Trump” movement to delegitimize his victory last November.

His focus on what he asserts was the inadequacy of Mr. Obama’s response to the hacking is new. He started sounding the theme last week and continued it on Monday: “The reason that President Obama did NOTHING about Russia after being notified by the CIA of meddling is that he expected Clinton would win.”

Completing the thought in his next tweet, Mr. Trump said: “…and did not want to ‘rock the boat.’ He didn’t ‘choke,’ he colluded or obstructed, and it did the Dems and Crooked Hillary no good.”

Mr. Trump made that assertion despite his own indifference to the hacking during the 2016 campaign, when at one point he said that he hoped Russian intelligence services had successfully hacked Hillary Clinton’s email, and encouraged them to publish whatever they may have stolen.

Just before 9 a.m. on Monday, he returned to the Obama-bungled-Russia theme, this time apparently referring to a Washington Post article chronicling confusion and delays by Mr. Obama’s aides when confronted with evidence that Moscow was attempting to intervene in the election.

“The real story is that President Obama did NOTHING after being informed in August about Russian meddling. With 4 months looking at Russia,” Mr. Trump wrote, and then added, “…under a magnifying glass, they have zero ‘tapes’ of T people colluding. There is no collusion & no obstruction. I should be given apology!”

Mr. Trump and some surrogates made the Post article the basis of their main talking point last week. Yet, as the president’s critics have pointed out, his tweets represent a tacit admission, for the first time, that Russia did attempt to undermine Mrs. Clinton through hacking and the dissemination of fake news on social media.

The special counsel, Robert S. Mueller III, and several congressional committees are investigating whether anyone on Mr. Trump’s campaign, or anyone working on his behalf, collaborated in that effort.

The president’s four tweets were bracketed by two other broadsides attacking Democrats for opposing legislative efforts by Republicans to roll back central components of the Affordable Care Act and replace it with their own health plan.

“The Democrats have become nothing but OBSTRUCTIONISTS, they have no policies or ideas. All they do is delay and complain. They own Obama Care!” Mr. Trump wrote at 8:30 a.m.

An hour later, after he venting over the Russia investigation, he added: “Republican Senators are working very hard to get there, with no help from the Democrats. Not easy! Perhaps just let Care crash & burn!”