New York City’s longest-serving congressman won’t pay his rent.

State taxpayers were stiffed out of at least $87,000 when Rep. Charles Rangel stopped paying for the district office he rents in Harlem’s Adam Clayton Powell Jr. State Office Building, records ­obtained by The Post show.

His staffers’ excuse? They lost the lease, according to state Office of General Services correspondence.

“I finally heard back from Congressman Rangel’s office and it seems we haven’t gotten the signed lease back because they lost it!” OGS real-estate specialist Sydney Allen wrote in a July 30, 2013, e-mail to a colleague that was ­obtained by The Post.

Rangel paid $7,253 in monthly rent on the 125th Street office he has rented since 2000, expense reports from 2012 show. But the payments stopped for all of 2013.

Incredibly, instead of demanding payment of the back rent and late fees from its deadbeat legislative tenant, the state cut him a huge rent break.

The state says it allowed Rangel in March 2013 to enter into a new sweetheart deal in which he could postpone paying six months of rent. That “abatement” money has still not been paid, nor has the other six months of missed rent from 2013, a OGS official said.

The state comptroller approved a $101,000 lease between Rangel and OGS on Dec. 26, 2013, retroactively covering the period back to April 2013 and future months through December 2014, records show. The 21-month deal resulted in a deeply reduced rent of $4,809 a month.

When The Post last week inquired about the year of missing rent, Rangel’s office and OGS blamed federal sequestration, not the lost lease referenced in agency correspondence.

“As everyone knows, the GOP sequester not only constrained our nation’s economy, but also strained the budget of congressional offices,” said Rangel spokeswoman Hannah Kim.

The sequester, which set caps on lawmakers’ discretionary spending, began in March 2013, two months after records show Rangel stopped paying rent.

Rangel also promised to open another district office in The Bronx, a portion of which he now represents after a 2012 redistricting, but still has not done so.

Rangel is no stranger to real-estate problems. Congress censured the 22-term legislator in 2010 after the Ethics Committee found him guilty of 11 violations — including dodging taxes on the rental of his Dominican Republic villa.

The then-chairman of the House Ways and Means Committee — he lost the powerful post due to his censure — was also forced to request an ethics probe into himself for his campaign’s use of one of four rent-controlled Harlem apartments that Rangel rented. Financial-disclosure forms show Rangel was worth between $566,000 and $1.2 million at the time.

The veteran Harlem pol is facing what could be his biggest political challenge. On June 24, primary voters will choose among him and two Democratic challengers — state Sen. Adriano Espaillat and the Rev. Michael Walrond.