As their Sweet 16 parties approach in the NCAA men’s basketball tournament, leaders of Virginia Commonwealth University and the University of Richmond are learning the same lesson as Butler University last year and George Mason University four years before that: You can’t buy this kind of publicity.

In the business of running a university, there is nothing quite so transformative as a long run in the annual intercollegiate athletic tournament known as March Madness.

All of a sudden, “VCU” is a hot search term on Google. The University of Richmond campus bookstore is unloading truckloads of Sweet 16 T-shirts. The neighboring schools — one a private liberal arts school, the other an under-funded state university — are the toast of Richmond.

A string of NCAA victories can accomplish more in a month than a decade’s worth of promotional mailings and campaigns to build brand identity. Any upstart entrant captures a small share of the nation’s rapt attention. Progress to the Sweet 16 or the Final Four, and that attention is divided fewer ways.

“It’s literally incalculable,” said Edward Ayers, president of the University of Richmond.

The ride could soon end for Richmond, which plays Kansas on Friday evening, and VCU, which plays Florida State later that same night. But for a measure of the benefit of a tournament run, consider Butler, a 4,500-student private university in Indiana. It made it to the tournament’s final game last year before losing to Duke. This year, applications to Butler are up 41 percent, the largest single-year bump in memory. Requests for campus visits spiked 35 percent.

“Those are the best two weeks of my life that happened last year,” said Tom Weede, vice president for enrollment management at Butler. “And I think most of the people on campus would answer the same way.”

Most universities that get to play for NCAA championships are big-name schools already, like Duke, the University of North Carolina and Georgetown. Intercollegiate glory adds visibility, but those schools already enjoy a national reputation.

For colleges like Butler and the two Richmond schools, March Madness means something altogether different. It’s a chance to gain a national reputation almost literally overnight.

The University of Richmond is a first-tier liberal arts school with selective admissions and an unusually large endowment, which permits the school to meet the full financial need of its students. Yet, the institution is not particularly well-known outside the region — except to those who study collegiate rankings.

“What we’re looking forward to is people hearing our story, not just getting the name out there,” said Ayers.

VCU, meanwhile, is a state university with a strong academic reputation but little visibility outside the state. But that may be changing. The university’s name ranked as a top-five “hot” search term on Google over the weekend, officials said.

“This week is a big week, and I can’t measure too many things, but one thing I can measure is Google hits,” said Michael Rao, the president.

“Our strategic plan was already to increase our outreach. This will help with that,” he added. “Our strategic plan was already to increase alumni giving. This will help with that.”

Last year, Butler benefited both from its staying power in the tournament and from a fortuitous narrative: Word got out that members of its basketball team had gone to class on a game day. Butler students, it seemed, always put their academics first. Hundreds of news organizations picked up the account.

“We could never have come up with a story like that,” Weede said.

George Mason’s Final Four run in 2006 proved such a potent Cinderella story that the school’s researchers mark that date as a before-and-after point. Since then, the school’s admission rate has improved from 70 percent to just over 50 percent.

As a state school with a relatively short history, George Mason simply didn’t register with much of the nation before its NCAA victories — it was just another mid-Atlantic school named for a lesser-known patriot.

“It gets you attention,” said Alan Merten, the president. “If you’re doing things wrong, it gets you negative attention. If you’re doing things right, it gets you positive attention.”

Adele McClure, student government president at VCU, has already seen signs at her school of the sort of electric spirit Merten saw five years ago at George Mason. More students are wearing school colors and VCU sweat shirts, and cashing in on their bragging rights.

“Our school spirit, it was good but it wasn’t great,” she said. “And now it’s beyond great.”