cp's OEIS Frontend

A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

%I A345267 #17 Jun 26 2025 16:31:52
%S A345267 1,2,2,48,1,1,1,240,1,2,2,1008,1,1,1,480,1,2,2,528,1,1,691,65520,1,2,
%T A345267 2,48,1,1,3617,16320,1,2,87734,57456,1,1,174611,13200,1,2,155366,1104,
%U A345267 1,1,236364091,131040,1,2,1315862,48,1,1,3392780147,6960,1,2
%N A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).
%C A345267 a(n) is known for n != 0 mod 4 and is related to the Bernoulli numbers via the Riemann zeta function. See Section VI.9 of Weibel's K-book.
%C A345267 K_0(Z) = Z is classical, so a(0) = 1. Rognes proved that K_4(Z) = 0 in 2000 so a(4) = 1.
%C A345267 Otherwise the value of a(4i) = 1 is conjectural. Kurihara observed that this follows from the Kummer-Vandiver conjecture (and in fact is equivalent with it). The Kummer-Vandiver conjecture has been verified for primes up to 163 million, from which it follows that a(4i) must be at least that large if it is not 1.
%D A345267 C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.
%H A345267 M. Kurihara, <a href="http://www.numdam.org/item/CM_1992__81_2_223_0/">Some remarks on conjectures about cyclotomic fields and K-groups of Z</a>, Compositio Math. 81 (1992), 223-236.
%H A345267 J. Rognes, <a href="https://doi.org/10.1016/S0040-9383(99)00007-5">K_4(Z) is the trivial group</a>, Topology 30 (2000), 267-281.
%H A345267 C. Weibel, <a href="https://sites.math.rutgers.edu/~weibel/Kbook.html">The K-book: An Introduction to Algebraic K-theory</a>. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.
%F A345267 Let k be the integer part of 1 + n/4.
%F A345267 a(n) is:
%F A345267               1 if n = 0 mod 8
%F A345267               2 if n = 1 mod 8
%F A345267    2*A000367(k) if n = 2 mod 8
%F A345267    2*A006863(k) if n = 3 mod 8
%F A345267               1 if n = 4 mod 8
%F A345267               1 if n = 5 mod 8
%F A345267   -1*A000367(k) if n = 6 mod 8
%F A345267      A006863(k) if n = 7 mod 8.
%o A345267 (Python)
%o A345267 from sympy import bernoulli
%o A345267 def a(n):
%o A345267     n_ = n % 8
%o A345267     k = n//4 + 1
%o A345267     b = bernoulli(2*k)/(4*k)
%o A345267     d = {0:1, 1:2, 4:1, 5:1}
%o A345267     if n_ == 2:
%o A345267         return 2 * b.numerator
%o A345267     elif n_ == 3:
%o A345267         return 2 * b.denominator
%o A345267     elif n_ == 6:
%o A345267         return -1 * b.numerator
%o A345267     elif n_ == 7:
%o A345267         return b.denominator
%o A345267     else:
%o A345267         return d[n_]
%Y A345267 Cf. A345225 (the order of the 2-primary subgroup, divides a(n)).
%Y A345267 A000367 / A006863 (numerator / denominator of B_2n/4n, where B_m are the Bernoulli numbers).
%K A345267 nonn,easy
%O A345267 0,2
%A A345267 _Tom Harris_, Jun 12 2021