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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).

[Edit (Jan. 5, 2021) An inspiring introductory talk on this topic was given by Michael Hopkins, "Bernoulli numbers, homotopy groups, and Milnor" last Feb.]

[Edit (May 22, 2021) A survey article containing recent computations: "Stable homotopy groups of spheres" Isaksen, Wang, and Xu.)

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.)

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    $\begingroup$ Stated more simply, those are the orientation-preserving diffeomorphism classes of homotopy spheres. As to why the numbers crop up, wouldn't the best resource be Kervaire-Milnor? IMO trying to view this through the lens of number theory isn't very fruitful. Numbers are part of the descriptive language but that doesn't mean they're really orchestrating the result. $\endgroup$ – Ryan Budney Oct 2 '15 at 22:04
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    $\begingroup$ Really? I think it is not uncommon. Mathematics is a large field with many people in it doing different things. $\endgroup$ – Sean Tilson Oct 5 '15 at 23:19
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    $\begingroup$ I am not aware of the History, but I might call your question "too discussiony" and I don't know that I would mean it in an insulting way. $\endgroup$ – Sean Tilson Oct 6 '15 at 12:01
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    $\begingroup$ I remember Richard Stanley once posed something like the following problem: "Fill in the missing number: 1, 1, 1, _, 1, 1, 1, ..." The answer is $\infty$, because this is the sequence of the number of differential structures on $\mathbb{R}^n$. $\endgroup$ – Sam Hopkins Nov 27 '16 at 16:58
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    $\begingroup$ @TomCopeland: I guess my point was that viewing these results in terms of sequences of numbers might not be so useful... $\endgroup$ – Sam Hopkins Nov 27 '16 at 18:14

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