The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as

$$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; 2,\; 8,\; 6,\; 992,\; 1,\; 3,\; 2,\; 16256, \;2,\; 16,\; 16,\; 523264,\; 24, \;8, \;4,\;...$$

with the numbers for the dimensions $4n+3$ greater than three sticking out like sore thumbs. It also juxtaposes the factors of the associated Kervaire-Milnor formula (described also enthusiastically by Mazur in "Bernoulli numbers and the unity of mathematics" by Barry Mazur, Secs. 4, 5, and 6).

Why do these particular dimensions stand out? What is the ultimate source of this peculiarity?

(Further notes are available at my website, including references by experts on connections to the Bernoulli numbers--Kervaire and Milnor.)