They are fanning out across the nation this weekend like a band of gypsies - begging a place to sleep from friends or relatives, renting motel and hotel rooms and, in one instance, camping out.

They are members of the United States Senate, the nation's most exclusive club. But when they leave Washington for the states that sent them here, many simply don't have a place to call their own.

Fifteen of the 100 senators do not own houses in the state they represent.

Two others own houses here and rent places back home.

For some senators, such as Pete V. Domenici (R-N.M.) and Carl Levin (D-Mich.), it's a matter of money. It was just too expensive to buy a house here and maintain a second property in their home states. Levin rents an apartment in Detroit and Domenici stays with his mother in Albuquerque.

Other senators have special circumstances. Herman E. Talmadge (D-Ga.) lost his place in Lovejoy as the result of his highly publicized divorce. An aide said Talmadge now stays with his son in Jonesboro when he goes home.

For freshman Republicans William L. Armstrong of Colorado and Richard G. Lugar of Indiana, trying to keep tenants in their houses was too much of a hassle.

Another 18 sentators own houses in their home states but either can't use them because they are rented or must share them with relatives or friends who occupy them year-round.

Max Baucus (D-Mont.) managed to find a tenant who moves out of his house in Missoula every time the Senate has a long recess, such as the month-long break that began Saturday.

An aide to Larry Pressler (R-S.D.), explaining why her boss stays with his parents on their farm near Humboldt, said "he's single, and he's only 37." As with several others, Pressler is "looking for a place" to buy in his home state.

Property in the Coyote state must be at a premium. An aide to George McGovern (D-S.D.) also said her boss is "looking for another house." McGovern sold his former home but still owns land in South Dakota.

William Proxmire (D-Wis.) and his staff sounded confused about whether he owned a house back home. Three aides insisted that the senator "lives in Madison." Proxmire then got on the telephone and said "yes, I have had a house in Madison since 1949. But I haven't lived in it since 1957," the year he first was elected to the Senate.

Asked if the house is rented, Proxmire said, "No, I'm selling it on contract."

Next question: Where is your legal residence? Answer: "I handle that."

Then Proximere added, "There's no law that says you have to live in the state you represent. I could run from Maryland, or Hawaii."

In fact, the U.S. Constitution says: "No person shall be a senator...who shall not, when elected, be an inhabitant of that state for which he shall be chosen." He may, however, move after the election.

Jennings Randolph (D-W.Va) owned an apartment in Elkins, but it was destroyed in a fire. Now his official residence is a room at the Tygart Hotel in Elkins.

"He pays for the room," an aide said, "but only for the nights he uses it."

Senate Majority Leader Robert C. Byrd (D-W. Va.) lists Sophia as his home, but he lives in Washington. "He sort of commutes to West Virginia," an aide said, nothing that Byrd has 18 trips scheduled to the Mountain State during the current recess.

"Frustration," said Colorado's Armstrong. Tenants wrecked his house in Aurora, and "it wasn't worth it to commute 1,700 miles to see that the lawn was watered."

Indiana's Lugar also cited "the inability to give attention to the maintenance" to his large house in Indianapolis. "It began to be a considerable burden," he said.

Sale of the house raised the question of Lugar's voting residence, but the problem was solved by the state board of elections, which ruled that the former Indianapolis mayor was "called away to national service," and his official address should remain the house that he sold.

When Lugar visits Indianapolis, he either stays with his mother in her apartment, or takes a room at a motel at the airport "because I'm always on the move anyway."

Another freshman, David L. Boren (D-Okla.), hasn't had to worry about a place to hang his hat back home for several years. Boren, 38, was born in Washington, graduated from Bethesda Chevy Chase High School and Yale University and was out of college only seven years when in 1975, he was elected governor and given a mansion to live in. His legal address, a post office box, is in Seminole, which he represented in the Oklahoma legislature.

Other senators who do not own houses in their home states are Thad (R-Miss.), who stays with his parents in Byram; William S. Cohen (R-Maine), who owns a farm with no house on it; Paul Laxalt (R-Nev.), who stays in motels when visiting his constituents; Gaylord Nelson (D-Wis.) and Bob Packwood (R-Ore.).

Senators Rudy Boschwitz (R-Minn.), Dennis DeConcini (D-Ariz.), John Emlcher (D-Mont.) and Robert T. Stafford (R-Vt.) own homes that are rented or leased on a long-term basis.

Stafford can stay with his sister or friends but chooses, for part of the current recess, to camp out in the Green Mountains of his native state.

Harrison Williams (D-N.J.) opted to sell his home in New Jersey and put his money into a house in Georgetown. He rents a cottage year-round in Bedminster, N.J.

Edmund S. Muskie (D-Me.) owns a house in Kennebunk Beach, but it doesn't have central heating, so Muskie stays with friends when he visits during Maine's long winters.

Three senators are in circumstances opposite those of their colleagues previously mentioned. Joseph R. Biden (D-Del.) and William V. Roth Jr. (R-Del.) try to commute daily by train from Wilmington, although Roth also owns a house here. And Paul S. Sarbanes (D-Md.) is driven to the Capitol each day from his home in Baltimore. Maryland's senior senator, Republican Charles McC. Mathias, needs only one house, in Chevy Chase.

Virginia's senators own homes in the District but also own property in the Old Dominion. Harry F. Byrd Jr. has a place in Winchester, and John W. Warner owns a 3,000-acre farm at Atoka, in Fauquier County.

While most senators move their families with them to Washington, the family of Robert Morgan (D-N.C.) lives in a subdivision house in Lillington and the teenaged son of Nancy Kasselbaum (R-Kan.) stayed in Wichita to finish high school.

Others senators send their families to the home states for the summer and join them there on weekends.

But for some, Washington is home, and the families remain here while the elected senators fly off for weekend meetings with constituents and then hurry back to Georgetown, Capital Hill, McLean, Potomac or Chevy Chase.

Such is the case with Lugar, who has two sons in college in Indiana who are coming to McLean this month for the family vacation.

While the reality is that most senators consider Washington home, for tax purposes it's a temporary abode. Even those senators who have no other home can claim a $3,000 annual tax deduction for the presumed extra expense of living here.

Section 162 of the Internal Revenue Code allows that deduction for persons "who in the pursuit of trade or business" must maintain a place of residence other than their primary one. An IRS spokesman said the section has been interpreted to apply to members of Congress. She added that "they don't have to determine where their tax home is," and they don't have to demonstrate that they have more than one residence.

The deduction not only survived the tax revisions of 1976 and 1978, but this year Congress is weighing a proposal that would set the deduction at $50 for every day Congress meets, rather than at the flat $3,000 limit. Last year when Congress met for 149 days, the deduction would have amounted to $7,450 against the annual salary of $57,500 for members.

Also contributing to this story was Washington Post researcher Regina Fraind. CAPTION: Picture 1, COCHRAN; Picture 2, PACKWOOD; Picture 3, PRESSLER; Picture 4, LUGAR; Picture 5, Cochran ARMSTRONG; Picture 6, McGOVERN; Picture 7, TALMADGE; Picture 8, BYRD; Picture 9, NELSON; Picture 10, RANDOLPH; Picture 11, BOREN; Picture 12, PROXMIRE; Picture 13, COHEN; Picture 14, LAXALT; Picture 15, DOMENICI