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a(n) = order of 2-primary subgroup of the group K_n(Z).
2

%I #28 Jun 13 2021 03:18:02

%S 1,2,2,16,1,1,1,16,1,2,2,16,1,1,1,32,1,2,2,16,1,1,1,48,1,2,2,16,1,1,1,

%T 64,1,2,2,16,1,1,1,80,1,2,2,16,1,1,1,96,1,2,2,16,1,1,1,112,1,2,2,16,1,

%U 1,1,128,1,2,2,16,1,1,1,144,1,2,2,16,1,1,1,160

%N a(n) = order of 2-primary subgroup of the group K_n(Z).

%C The algebraic K-theory groups of the integers are not yet (as of June 2021) completely known, however the 2-primary part is settled.

%C In fact, the 2-primary part of the abelian group K_n(Z) is the cyclic group Z/a(n)Z.

%D C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-9132-.

%H Tom Harris, <a href="/A345225/b345225.txt">Table of n, a(n) for n = 0..9999</a>

%H C. Weibel, <a href="https://doi.org/10.1016/S0764-4442(97)86977-7">The 2-torsion in the K-theory of the integers</a>, C. R. Acad. Sci. Paris 324 (1997), 615-620.

%F a(n) is:

%F 1 if n mod 8 = 0, 4, 5, or 6

%F 2 if n mod 8 = 1 or 2

%F 16 if n mod 8 = 3

%F 2*(n+1) if n mod 8 = 7.

%F (The main result of Weibel's 1997 paper on the 2-torsion in the K-theory of the integers; Corollary 9.8 of Weibel's K-book.)

%o (Python)

%o def a(n):

%o n_ = (n % 8)

%o d = {0:1, 1:2, 2:2, 3:16, 4:1, 5:1, 6:1}

%o if n_ == 7:

%o return 2*(n+1)

%o else:

%o return d[n_]

%K easy,nonn

%O 0,2

%A _Tom Harris_, Jun 11 2021