A345225 a(n) = order of 2-primary subgroup of the group K_n(Z).

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, 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, 1, 1, 128, 1, 2, 2, 16, 1, 1, 1, 144, 1, 2, 2, 16, 1, 1, 1, 160

Offset: 0

Author

Tom Harris, Jun 11 2021

Keywords

• easy
• nonn

Comments

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

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

References

• 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-.

Links

• Tom Harris, Table of n, a(n) for n = 0..9999
• C. Weibel, The 2-torsion in the K-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615-620.

Programs

Python

def a(n):
    n_ = (n % 8)
    d = {0:1, 1:2, 2:2, 3:16, 4:1, 5:1, 6:1}
    if n_ == 7:
        return 2*(n+1)
    else:
        return d[n_]

Formula

a(n) is:

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

2 if n mod 8 = 1 or 2

16 if n mod 8 = 3

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

(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.)