ALBANY – More than 37,000 New Yorkers have signed up for health insurance since enrollment under the Affordable Care Act began Oct. 1, but many of them are going into the state’s Medicaid program rather than one of 16 private insurance carriers participating in the new federally mandated program.

State officials and private insurance executives say they anticipated the higher Medicaid enrollment under the health care law, and the state budget plan enacted this year projects higher federal reimbursement rates to make up for more people moving into Medicaid.

In the first three weeks of the “Obamacare” enrollment, 23,717 New Yorkers who went to a new state website to enroll for insurance ended up being steered to join Medicaid because their incomes were low enough to qualify.

In all, 13,313 New Yorkers have enrolled for health coverage with one of the 16 private insurance carriers participating. While the state has begun giving carriers overall numbers, insurance industry officials cautioned that few of the 13,313 have actually gone through the final process with the companies to become officially enrolled.

Insurance companies also say the major technological glitches have been resolved. In the early few days of the program, the state’s website crashed often as people could not get onto the page to shop or sign up for insurance. That problem is still affecting the larger, federally administered system being used to enroll people in 36 states without their own sign-up system.

Leslie S. Moran, senior vice president at the New York Health Plan Association, which represents many of the private companies involved in the state’s program under the health care law, said the state and industry have been having regular teleconferences to settle issues that have arisen.

“We’ve had a real good relationship and therefore an ability to flag problems very quickly and work to make sure they get fixed very quickly,” she said.

Moran said the enrollment numbers provided by the state are still a work in progress. “It doesn’t mean a bill has been generated for a premium or that we’ve captured that person and enrolled them yet. But for all intents and purposes, 37,000 New Yorkers have gone on the website and done more than shop. They’ve selected a plan in which to enroll,” she said.

The state’s Medicaid program costs $55.7 billion, including the money counties pay into the system. The state’s enacted budget plan from Gov. Andrew M. Cuomo’s fiscal office projects Medicaid enrollment to rise by 7.3 percent, to 5.6 million people, by 2014. The budget document also anticipates the federal government picking up a higher portion of Medicaid expenses because officials knew that more people were likely to end up going into that program rather than one through an insurance exchange.

Moran said that many of those new Medicaid enrollees who thought they were signing up for one of the private insurance plans under “Obamacare” are an easier population to reach with word about the new system because they likely are already receiving some sort of assistance, such as food stamps.

“Part of the Affordable Care Act was to expand eligibility of Medicaid and to capture people eligible for Medicaid but who were not already in that program,” she said.

The state has projected that 1.1 million people will get insurance under “Obamacare” in the first three years.

State health officials say 174,000 New Yorkers have completed an application for insurance. But that does not mean all will end up being enrolled.