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

0,2

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.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

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.)

G.f.: (1 + 2*x + 2*x^2 + 16*x^3 + x^4 + x^5 + x^6 + 16*x^7 - x^8 - 2*x^9 - 2*x^10 - 16*x^11 - x^12 - x^13 - x^14)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2*(1 + x^4)^2). - Andrew Howroyd, Nov 11 2025

Prog

(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_]

Crossrefs

Sequence in context: A184718 A079897 A097540 * A373997 A112327 A152541

Adjacent sequences: A345222 A345223 A345224 * A345226 A345227 A345228

Keyword

easy,nonn

Author

Tom Harris, Jun 11 2021

Status

approved