Dan Quayle is heading overseas this week, to realms where vice presidents are taken seriously.

It is his 13th foreign trip as vice president, and this one will be a respite from the wisecracks and qualms that pushed the Quayle factor up the political agenda after President Bush was hospitalized with an irregular heartbeat.At home, most vice presidential assignments are as secondary as the job description. When there is important work, it doesn't draw much attention because it has to be conducted in a supporting role.

Abroad, a visiting American vice president stars in the main event. It's not quite presidential treatment, but it's not far short.

Quayle's latest trip takes him to Japan, Singapore and Indonesia. He is to return to Washington on Friday. There's another foreign assignment coming next month, June 2 to 7, in eastern Europe. When that one is completed, he will have visited 36 countries during his 21/2 years as vice president.

Foreign travel, including but not limited to the funeral duty Bush used to joke about when he was vice president, has become an increasingly important part of the job in the past 40 years. The emphasis dates from Richard Nixon's two terms; he traveled to 54 countries. His Moscow "kitchen debate" with Nikita Khrushchev in 1959 was a political bonus at home.

Quayle could use a boost like that. The uproar over his role, now and on the 1992 Republican ticket, has subsided now that Bush's heart irregularity has been diagnosed, controlled and is under treatment.

But the clamor previewed a political argument that is guaranteed a rerun when the 1992 campaign begins. Bush has said Quayle will again be on his ticket.

"I think it's an issue regardless of what we say or don't say about him," Sen. George J. Mitchell, the Democratic leader, said in a television interview. "It's obvious the American people have deep concerns about it, and I think it's an appropriate consideration. . . . "

Democrats looking for Bush's vulnerabilities certainly will make it one. But Quayle also has had Democratic defenders, notably House Speaker Thomas Foley.

"I don't have the slightest concern about Vice President Quayle's ability to carry on the duties of his office, which include any possible action of transfer of authority under the 25th Amendment," Foley said.

"I wish everybody would just lay off Dan Quayle," added Sen. Patrick Leahy, D-Vt. "At some point it goes way beyond fairness."

Bush said Quayle has been taking a bum rap while doing a first-class job.

"I said, `Look, keep your head up. You're talking to a guy that went through something like this for about eight years, maybe not quite as intense.'

"And it is unfair, and it is piling on, and it is beneath the critics to do that," he said.

That shared experience is one of the reasons Bush will resist any change in his ticket.

Quayle said all he can do is plow ahead, no matter what the critics say. "This is not a high-profile office nor does this office give you the ability to change perceptions dramatically," he told USA Today.

He meets with Bush every morning when they're both in town. He's worked at overhauling space policy and tailoring it to a tight budget. His council on U.S. competitiveness has revived an administration in which the benefit of environmental and other regulations have to be weighed against the cost, and some have been scrapped as a result. Quayle "has had more impact on foreign and domestic policy than any vice president, including George Bush," says Secretary of Transportation Samuel Skinner.

But that doesn't register in the public opinion polls.

And there's no letup in the wisecracks. Comic Jay Leno says Quayle was standing close by, all through Bush's illness - in case there was need of a heart trans-plant.

- Walter Mears is vice president and columnist for The Associated Press.