Looking at Kervaire and Milnor's^{1} classification of exotic spheres, for each $n$ there are a finite number of distinct (up to diffeomorphism) smooth structures you can impose on the $n$-sphere (OEIS sequence A001676). The first few terms of this sequence, denoted $[\Theta_n]$, are

$$ \begin{array}{c|cc} n & 1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&\dotsb\\\hline [\Theta_n] & 1&1&1&\geq1&1&1&28&2&8&6&992&1&3&2&16256&\dotsb\\ \end{array} \;.$$

There are obvious spikes in the value of $[\Theta_n]$ for $n \equiv 3 \!\pmod 4$. Is there a nice reason why this happens? The details of the linked paper are a bit over my head.

Wikipedia gives a brief explanation: $\Theta_n$ is the group of $h$-cobordism classes of the homotopy $n$-sphere, and there's a correspondence between the elements of this group and the number of smooth structures of the $n$-sphere. Each group $\Theta_n$ contains a cyclic subgroup $bP_{n+1}$ of the "$n$-spheres that bound parallelizable manifolds." And then for some reason the size of $bP_{n+1}$ can be very large for $n \equiv 3 \!\pmod 4$ and somehow relates to Bernoulli numbers.

- Kervaire and Milnor, Groups of homotopy spheres: I, Ann. Math. (2) 77 (1963), 504-537

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