The outgoing comptroller general of the United States has sent a ter letter to Defense Secretary Caspar W. Weinberger declaring that mismanagement and waste are now adding "several billion dollars a year" to the Pentagon budget, which the Reagan administration proposes to increase dramatically.

The letter, sent to Weinberger on his first day in office by Elmer B. Staats, head of the General Accounting Office, listed 15 major areas where large amounts of money could be readily saved. Staats called them "some of the most apparent high payoff opportunities" in the defense budget.

GAO has 1,000 accountants and other professional continuously studying Pentagon programs, and Staats' letter was based on repeated audits they have made. Weinberger has yet to respond to the letter in detail. President Reagan vowed to root out mismanagement and waste throughout the government during last year's campaign.

Just the 15 area cited have the potential to save an "absolute minimum" of $4 billion a year by the end of Reagan's term in 1985, Staats said in an interview last week. If implemented, other Department of Defense (DOD) economies urged by the GAO would increase annual savings by 1985 to "substantially more" than $10 billion, Staats said.

Staats' estimates come as the administration is preparing to increase the defense budget by billions of dollars next year, while cutting back on domestic programs. The administration has said it will seek to hold down the defense increase by finding savings wherever it can in the Pentagon. But, so far, it has not specified any.

Staats' proposed savings do not challenge the basic defense policies laid down by Congress and the Joint Chiefs of Staff.

Among the major points made by Staats in his letter and the interview:

Thirty GAO reports since 1976 have cited serious flaws in the way DOD accounts for goods and services provided under the foreign military sales program. These have resulted in "improperly" made payments of more than $1 billion. Some progress has been made, but "more improvement is needed."

At the outset of a new weapons system, the Pentagon often tries to hold down the apparent total cost by downplaying the need for the spare parts and back-up equipment needed over the long term. This means that the parts and equipment must be bought later, at greater cost. In the case of the F18 fighter, the savings potential in proper logistical planning and discipline throughout its life cycle would be more than $350 million.

In many geographic areas, the Army, Navy, Air Force, and Marines each have one or more bases. Instead of sharing such things as repairs and supplies, each of the armed forces provides them separately. The potential savings from consolidation: at least $376 million annually.

Management of consumable items, such as ammunition and motor vehicle parts, is scattered. Consolidation would yield yearly savings exceeding $280 million.

In the last 16 months, the estimated cost of the Navy's LAMPS III (light airborne multipurpose antisubmarine weapon system) has almost doubled, from $3.6 billion to $7 billion. Now, the plan is to buy fewer of them. But that does not solve the questions raised by a GAO review "about the ability of the system to carry out its missions."

DOD plans to buy large numbers of F14 F15, F16, F18 and A10 aircraft for noncombat purposes, asserting that they would be needed in event of war. Staats said that such planning was "very sloppy" and claimed that sharpening it up could save up to $5 billion.

"A hard look at this area by the secretary of defense is needed," he said. "If these aircraft are in fact required for combat purposes, they should be justified as such and approved in that context by the secretary, the OMB and the Congress."

In 1972, the Army got started on its supertank, the M1. The cost was to be $507,000 a copy. By January 1980, when a GAO report warned that the tank's reliability was "uncertain," the unit cost had more than tripled to $1,651,000. The number of tanks in the program was 7,058.

Now the estimated unit cost has shot up to $2,549,000. Meanwhile, the GAO is still reviewing reliability and other aspects of the M1.

Four weeks after the Staats letter went out, the Pentagon issued its quarterly report on the life cycle estimated costs of 47 selected major weapons systems. The figure for Dec. 31 was $47.6 billion higher than it has been only three months earlier. DOD cast inflation as the chief villain behind the increase, which over the three-month period was at the rate of $359,300 a minute.

Liberals in Congress have traditionally denounced the defense budget as excessive. Now, with costs rising dramatically, some conservatives have begun to seek further justification as well. Last Wednesday, an unusual strong joint plea for "decisive action" against defense waste went to Weinberger from Republican Sen. Barry Goldwater and liberal Democratic Sen. Howard M. Metzenbaum.

"Runaway costs characterize our entire defense procurement program," although they "have nothing to do with maintaining the strength of our military forces," the senators said in a letter to Weinberger. "They are, pure and simple, the result of a system that permits DOD officials to operate as though the public purse has no limits."

Saats, who had four top GAO defense specialists at his side during the interview, said the Pentagon has repeatedly ignored documented advice on how to eradicate mismanagement and waste. At the same time, he praised DOD for accepting GAO recommendations that in the last five years produced "measurable savings" of $8.9 billion -- 54 percent of the total for the entire government. The DOD share of the budget has been much smaller, about 24 percent.

Staats brings the prestige of his office and his long, nonpartisan government service to his plea for defense economies. He is retiring Friday after 15 years as comptroller general and will make his last public appearance before Congress today when he testifies before the House Budget Committee.

This is "the ideal time" to change direction at the Pentagon, he said in the interview.

He has received support from many sources. One, the Republican Study Group in the House, when it was headed by GOP conservative John H. Rousselot (Calif.) last August, released a summary of 19 months of GAO reports showing potential DOD savings of $16 billion.

Staats began immediately after the November election to alert key Reagan aides to governmentwide cost-cutting opportunities, sending what he called "vast amounts" of GAO materials to Edwin Meese III, counselor to the president; Weinberger, whose cost-cutting in earlier government service earned him the nickname "Cap the Knife," and budget director David A. Stockman.

Up to now, the Staats letter has elicited only an acknowledgement from the defense secretary. But, Staats and his aides said, there are "all kinds of indications" that the Pentagon is taking the letter "very seriously."

Many Pentagon critics share a hope that Weinberger will renounce attitudes of the kind held in the Carter administration by W. Graham Claytor, secretary of the Navy and then deputy secretary of defense. "I have said many times there is nothing wrong with the Navy that money won't fix," he once told the Senate Armed Services Committee.

In a speech in 1979, Gordon W. Rule, who had retired after being honored by the Navy as its most distinguished civilian procurement specialist, termed Claytor's view "reprehensible." In his own recent letter to Weinberger, he charged that the military is "almost devoid of demonstrable cost consciousness and accountability for mistakes or poor judgment decisions."

Staats, for whom salty talk like Reagan Rule's would be out of character, stops short of such drastic judgments. "The accountabiity system isn't as good as it ought to be," he said.

But he spoke plainly about what may happen if an unreformed Pentagon gets the five-year $169.5 billion spending boost proposed by Reagan.

Insofar as the funds go for new initiatives rather than purposes such as strengthening conventional forces, there "would be a problem," Staats said. "You just cannot absorb much [new] money very fast."

The administration's "Program for Economic Recovery," as announced two weeks ago, projected reductions in defense outlays in the four fiscal years 1981 through 1984 of $13.9 billion, including $5.9 billion in the final year. But the program listed almost no specifics, especially as compared with the detailing of plans to reduce outlays in nonmilitary areas by about $41.5 billion.