We last did this in 2018 (although Rockefeller University was incorrectly omitted last time).  The National Academy of Sciences is the even more selective "science only" cousin of the American Academy of Arts & Sciences. While it does elect faculty in the social sciences, its focus is on the natural sciences, engineering and other applied sciences, and the biological sciences, including those in and around medicine; any university with a good medical school tends to do better, all else being equal, in NAS representation (indeed, some universities with top medical schools, like UC San Francisco and Texas Southwestern in Dallas make the list, despite not being full service research universities). As a result I mark with an asterisk (*) universities without medical schools, and I mark with a # universities with medical schools with very high research profiles (Some schools, like UT Austin, UC Riverside and Illinois/Urbana have only recently added medical schools.) As with the AAAS, there is a certain amount of "the rich get richer" at the NAS, though the schools with the most members are indeed very strong in the sciences.

1.  #Harvard University (188)

2.  #Stanford University (164)

3.  *University of California, Berkeley (125)

4.  *Massachusetts Institute of Technology (120)

5.  *Princeton University (89)

6.  *California Institute of Technology (70)

7.  #Columbia University (69)

8.  Yale University (67)

9.  University of California, San Diego (53)

10. University of Chicago (48)

11. #University of California, San Francisco (46)

11. #University of Washington, Seattle (46)

13. New York University (45)

14. #University of California, Los Angeles (43)

15. #University of Pennsylvania (40)

16. Cornell University (34)

16. *Rockefeller University (34)

16. University of Michigan, Ann Arbor (34)

19. University of Wisconsin, Madison (33)

20. #Johns Hopkins University (32)

21. *University of California, Santa Barbara (30)

22. #Washington University, St. Louis (28)

23. #Duke University (27)

23. Northwestern University (27)

23. University of Illinois, Urbana-Champaign (27) (young medical school)

26. *Rutgers University, New Brunswick (24)

27. University of California, Davis (23)

27. #University of Texas Southwestern (Dallas) (23)

29. University of California, Irvine (22)

30. University of Colorado, Boulder (21)

31. University of Minnesota, Minneapolis-St. Paul (19)

32. *Scripps Research Institute (San Diego) (18)

32. University of Texas, Austin (18) (young medical school)

34. *Arizona State University (17)

34. Pennsylvania State University (17)

34. University of Florida, Gainesville (17)

37. University of Arizona (16)

38. University of Utah (15)

39. *University of California, Santa Cruz (14)

40. *University of Maryland, College Park (13)

40. University of North Carolina, Chapel Hill (13)

42. *Texas A&M University (12)

42. University of Southern California (12)

44. *North Carolina State University (11)

44. Ohio State University (11)

46. *Carnegie-Mellon University (10)

46. *Rice University (10)

46. University of California, Riverside (10) (young medical school)

49. Brown University (9)

49. Indiana University, Bloomington (9)

49. State University of New York, Stony Brook (9)

49. *University of Oregon (9)

Some other well-known schools (this list may not be complete, so is not an ordinal ranking):  Boston University (8); Brandeis University (8); University of Pittsburgh (7); Emory University (6); Vanderbilt University (6); Georgia Institute of Technology (4)