7
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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).

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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    $\begingroup$ Who keeps censoring my comments? $\endgroup$ – Tom Copeland Oct 3 '15 at 17:48
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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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    $\begingroup$ As to who is 'censoring' comments, the record is that I deleted one comment that had no math in it, and I edited some comments of both Ryan and Tom to remove some 'rude' or non-constructive content, in response to user flags. Tom has deleted four of his own comments. More discussion at MO meta: meta.mathoverflow.net/a/2511/2926 $\endgroup$ – Todd Trimble Oct 6 '15 at 14:26
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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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